One Hat Cyber Team

Edit File: K3Surfaces.m2

if version#"VERSION" < "1.18" then error "this package requires Macaulay2 version 1.18 or newer";

newPackage(
    "K3Surfaces",
    Version => "1.1", 
    Date => "August 13, 2022",
    Authors => {{Name => "Michael Hoff", 
                 Email => "hahn@math.uni-sb.de"},
                {Name => "Giovanni Staglianò", 
                 Email => "giovanni.stagliano@unict.it"}},
    PackageExports => {"SpecialFanoFourfolds"},
    Keywords => {"Algebraic Geometry"},
    Headline => "Explicit constructions of K3 surfaces",
    DebuggingMode => false
)

if SpecialFanoFourfolds.Options.Version < "2.6" then (
    <<endl<<"Your version of the SpecialFanoFourfolds package is outdated (required version 2.6 or newer);"<<endl;
    <<"you can manually download the latest version from"<<endl;
    <<"https://github.com/Macaulay2/M2/tree/development/M2/Macaulay2/packages."<<endl;
    <<"To automatically download the latest version of SpecialFanoFourfolds in your current directory,"<<endl;
    <<"you may run the following Macaulay2 code:"<<endl<<"***"<<endl<<endl;
    <<///run "curl -s -o SpecialFanoFourfolds.m2 https://raw.githubusercontent.com/Macaulay2/M2/development/M2/Macaulay2/packages/SpecialFanoFourfolds.m2";///<<endl<<endl<<"***"<<endl;
    error "required SpecialFanoFourfolds package version 2.6 or newer";
);

export{"K3","LatticePolarizedK3surface","EmbeddedK3surface","project","mukaiModel",
       "trigonalK3","tetragonalK3","pentagonalK3"}

debug SpecialFanoFourfolds;
needsPackage "Truncations";
needsPackage "MinimalPrimes";

LatticePolarizedK3surface = new Type of HashTable;

LatticePolarizedK3surface.synonym = "K3 surface";

EmbeddedK3surface = new Type of EmbeddedProjectiveVariety;

EmbeddedK3surface.synonym = "embedded K3 surface";

net LatticePolarizedK3surface := S -> (
    M := S#"lattice";
    d := M_(0,1);
    n := M_(1,1);
    g := lift((M_(0,0) + 2)/2,ZZ);
    w := "K3 surface with rank 2 lattice defined by the intersection matrix: "|net(M);
    local g'; local c; local a; local b; local g0; local d0; local n0;
    w = w || concatenate for s in select(fewPossibilities,l -> take(l,{4,6}) == (g,d,n)) list ( 
    (g',c,a,b,g0,d0,n0) = s;
    "-- "|toString(a,b)|": K3 surface of genus "|toString(g')|" and degree "|toString(2*g'-2)|" containing "|(if n == 0 then "elliptic" else "rational")|" curve of degree "|toString(c)|" "|(if isAdmissible(2*g'-2) then "(cubic fourfold) " else "")|(if isAdmissibleGM(2*g'-2) then "(GM fourfold) " else "")|newline
    )
);
texMath LatticePolarizedK3surface := texMath @@ net;

LatticePolarizedK3surface#{WebApp,AfterPrint} = LatticePolarizedK3surface#{WebApp,AfterNoPrint} = 
LatticePolarizedK3surface#{Standard,AfterPrint} = LatticePolarizedK3surface#{Standard,AfterNoPrint} = S -> (
    << endl << concatenate(interpreterDepth:"o") << lineNumber << " : " << "Lattice-polarized K3 surface" << endl;
);

EmbeddedK3surface#{WebApp,AfterPrint} = EmbeddedK3surface#{WebApp,AfterNoPrint} = 
EmbeddedK3surface#{Standard,AfterPrint} = EmbeddedK3surface#{Standard,AfterNoPrint} = S -> (
    << endl << concatenate(interpreterDepth:"o") << lineNumber << " : " << "Embedded K3 surface" << endl;
);

map (LatticePolarizedK3surface,ZZ,ZZ) := o -> (S,a,b) -> (
    if S.cache#?("map",a,b) then return S.cache#("map",a,b);
    M := S#"lattice";
    d := M_(0,1);
    n := M_(1,1);
    g := lift((M_(0,0) + 2)/2,ZZ);
    if d == 0 and n == -2 then if b != 0 then error "the K3 surface is nodal";
    H := hyperplane S;
    C := S#"curve";
    phi := mapDefinedByDivisor(Var S,{(H,a),(C,b)});
    if dim target phi =!= genus(S,a,b) then error("expected map to PP^"|(toString genus(S,a,b))|", but got map to PP^"|toString(dim target phi));
    S.cache#("map",a,b) = phi
);

map (EmbeddedK3surface,ZZ,ZZ) := o -> (S,a,b) -> map(K3 S,a,b);

map EmbeddedK3surface := o -> S -> (
    if not(S.cache#?"GeneralK3" and S.cache#"GeneralK3") then error "expected a general K3 surface of some genus";
    if not S.cache#?"mapK3" then error "construction map not found";
    S.cache#"mapK3"
);

coefficientRing LatticePolarizedK3surface := S -> coefficientRing Var S;

genus (LatticePolarizedK3surface,ZZ,ZZ) := (S,a,b) -> (
    M := S#"lattice";
    d := M_(0,1);
    n := M_(1,1);
    g := lift((M_(0,0) + 2)/2,ZZ);
    lift((a^2*(2*g-2) + 2*a*b*d + b^2*n + 2)/2,ZZ)
);

genus (EmbeddedK3surface,ZZ,ZZ) := (S,a,b) -> genus(K3 S,a,b);

genus EmbeddedK3surface := S -> sectionalGenus S;

degree (LatticePolarizedK3surface,ZZ,ZZ) := (S,a,b) -> 2 * genus(S,a,b) - 2;

degree (EmbeddedK3surface,ZZ,ZZ) := (S,a,b) -> degree(K3 S,a,b);

degree EmbeddedK3surface := S -> 2 * genus(S) - 2;

construction = method();
construction EmbeddedK3surface := (cacheValue "construction-L.P.K3") (T -> (
    if T.cache#?"GeneralK3" and T.cache#"GeneralK3" then error "expected K3 surface to be not general";
    error "unable to recover the construction for the embedded K3 surface";
));

LatticePolarizedK3surface Sequence := (S,ab) -> (
    if not(#ab == 2 and instance(first ab,ZZ) and instance(last ab,ZZ)) then error "expected a sequence of two integers";
    (a,b) := ab;
    if S.cache#?("var",a,b) then return S.cache#("var",a,b);    
    f := map(S,a,b);
    if f#"image" === null and char coefficientRing S <= 65521 and genus(S,1,0) > 3 then image(f,"F4");
    T := new EmbeddedK3surface from image f;
    if dim ambient T <= 2 then error "the linear system is not very ample";
    -- (???) this fixes a bug in conversion of output to net, but it is a bit dangerous.
    T#"dimVariety" = 2;
    T.cache#"sectionalGenus" = genus(S,a,b);
    -- if degrees T =!= {({2},binomial(genus(T)-2,2))} then <<"--warning: the degrees for the generators are not as expected"<<endl;
    f#"image" = T;
    T.cache#"construction-L.P.K3" = (S,ab);
    S.cache#("var",a,b) = T
);

EmbeddedK3surface Sequence := (S,ab) -> (K3 S) ab;

Var LatticePolarizedK3surface := o -> S -> S#"surface";

hyperplane = method();
hyperplane LatticePolarizedK3surface := (cacheValue "hyperplane") (T -> random(1,0_(Var T)));

latticePolarizedK3surface = method();
latticePolarizedK3surface (EmbeddedK3surface,EmbeddedProjectiveVariety,List) := (T,C,gdn) -> (
    (g,d,n) := toSequence gdn;
    new LatticePolarizedK3surface from {
        symbol cache => new CacheTable,
        "surface" => T,
        "curve" => C,
        "lattice" => matrix{{2*g-2,d},{d,n}}
    }
);
latticePolarizedK3surface (EmbeddedK3surface,EmbeddedProjectiveVariety,EmbeddedProjectiveVariety,List) := (T,C,H,gdn) -> (
    S := latticePolarizedK3surface(T,C,gdn);
    S.cache#"hyperplane" = H;
    return S;
);
latticePolarizedK3surface (EmbeddedK3surface,EmbeddedProjectiveVariety,Nothing,List) := (T,C,H,gdn) -> latticePolarizedK3surface(T,C,gdn);
latticePolarizedK3surface (EmbeddedProjectiveVariety,EmbeddedProjectiveVariety,List) := (T,C,gdn) -> latticePolarizedK3surface(new EmbeddedK3surface from T,C,gdn);
latticePolarizedK3surface (EmbeddedProjectiveVariety,EmbeddedProjectiveVariety,EmbeddedProjectiveVariety,List) := (T,C,H,gdn) -> latticePolarizedK3surface(new EmbeddedK3surface from T,C,H,gdn);
latticePolarizedK3surface (EmbeddedProjectiveVariety,EmbeddedProjectiveVariety,Nothing,List) := (T,C,H,gdn) -> latticePolarizedK3surface(new EmbeddedK3surface from T,C,H,gdn);

K3 = method(Options => {CoefficientRing => ZZ/65521, Verbose => false, Singular => null});

makegeneralK3 = (f,p,g) -> (
    K3surf := new EmbeddedK3surface from image f;
    assert(sectionalGenus K3surf == g and degree K3surf == 2*g-2 and dim ambient K3surf == g and dim p <= 0 and isSubset(p,K3surf));
    -- if g <= 12 then assert(degree p == 1);
    f#"image" = K3surf;
    K3surf.cache#"mapK3" = f;
    K3surf.cache#"pointK3" = p;
    K3surf.cache#"GeneralK3" = true;
    K3surf
);

K3 ZZ := o -> g -> (
    K := o.CoefficientRing;
    local X; local p; local Ass;
    if member(g,{3,4,5,6,7,8,9,10,12}) then (
        (X,p) = randomPointedMukaiThreefold(g,CoefficientRing=>K);
        j := parametrize random({1},p);
        X = j^* X; p = j^* p;
        return makegeneralK3(super 1_X,p,g);
    );
    if g == 11 then (
        if o.Verbose then <<"-- constructing general K3 surface of genus "<<g<<" and degree "<<2*g-2<<" in PP^"<<g<<endl;
        if o.Verbose then <<"-- (taking a random GM fourfold X of discriminant 20, hence containing a surface S of degree 9 and genus 2)"<<endl;
        X = specialGushelMukaiFourfold("general GM 4-fold of discriminant 20",K);
        if o.Verbose then <<"-- (running procedure 'associatedK3surface' for the GM fourfold X of discriminant 20)"<<endl<<"-- *** --"<<endl;
        Ass = building associatedK3surface(X,Verbose=>o.Verbose,Singular=>o.Singular);
        if o.Verbose then <<"-- *** --"<<endl;
        return makegeneralK3(last Ass,(last Ass) first Ass_2,g);
    );
    if g == 14 then (
        if o.Verbose then <<"-- constructing general K3 surface of genus "<<g<<" and degree "<<2*g-2<<" in PP^"<<g<<endl;
        if o.Verbose then <<"-- (taking a random cubic fourfold X of discriminant 26, hence containing a surface S of degree 7 and genus 1)"<<endl;
        X = specialCubicFourfold("one-nodal septic del Pezzo surface",K);
        if o.Verbose then <<"-- (running procedure 'associatedK3surface' for the cubic fourfold X of discriminant 26)"<<endl<<"-- *** --"<<endl;
        Ass = building associatedK3surface(X,Verbose=>o.Verbose,Singular=>o.Singular);
        if o.Verbose then <<"-- *** --"<<endl;
        return makegeneralK3(last Ass,(last Ass) first Ass_2,g);
    );    
    if g == 20 then (
        if o.Verbose then <<"-- constructing general K3 surface of genus "<<g<<" and degree "<<2*g-2<<" in PP^"<<g<<endl;
        if o.Verbose then <<"-- (taking a random cubic fourfold X of discriminant 38, hence containing a surface S of degree 10 and genus 6)"<<endl;
        X = specialCubicFourfold("C38",K);
        if o.Verbose then <<"-- (running procedure 'associatedK3surface' for the cubic fourfold X of discriminant 38)"<<endl<<"-- *** --"<<endl;
        Ass = building associatedK3surface(X,Verbose=>o.Verbose,Singular=>o.Singular);
        if o.Verbose then <<"-- *** --"<<endl;
        return makegeneralK3(last Ass,(last Ass) first Ass_2,g);
    );
    if g == 22 then (
        if o.Verbose then <<"-- constructing general K3 surface of genus "<<g<<" and degree "<<2*g-2<<" in PP^"<<g<<endl;
        if o.Verbose then <<"-- (taking a random cubic fourfold X of discriminant 42, hence containing a surface S of degree 9 and genus 2)"<<endl;
        X = specialCubicFourfold("C42",K);
        if o.Verbose then <<"-- (running procedure 'associatedK3surface' for the cubic fourfold X of discriminant 42)"<<endl<<"-- *** --"<<endl;
        Ass = building associatedK3surface(X,Verbose=>o.Verbose,Singular=>o.Singular);
        if o.Verbose then <<"-- *** --"<<endl;
        return makegeneralK3(last Ass,(last Ass) first Ass_2,g);
    );
    error ("no procedure found to construct random K3 surface of genus "|(toString g));
);

K3 SpecialGushelMukaiFourfold := K3 SpecialCubicFourfold := o -> X -> (
    d := discriminant X;
    if (not isAdmissible d) and (not isAdmissibleGM d) then <<"--warning: expected an admissible integer for the discriminant"<<endl;
    g := lift((d+2)/2,ZZ);
    if o.Verbose then <<"-- (running procedure 'associatedK3surface' for the fourfold of discriminant "<<d<<")"<<endl<<"-- *** --"<<endl;
    Ass := building associatedK3surface(X,Verbose=>o.Verbose,Singular=>o.Singular);
    if o.Verbose then <<"-- *** --"<<endl;
    return makegeneralK3(last Ass,(last Ass) first Ass_2,g);
);

K3 (ZZ,ZZ,ZZ) := o -> (g,d,n) -> (
    -- if o.Verbose then <<"-- constructing K3 surface with rank 2 lattice defined by the intersection matrix "<<matrix{{2*g-2,d},{d,n}}<<endl;   
    K := o.CoefficientRing;
    local X; local T; local C; local p; local j;
    if g >= 3 and g <= 12 and d == 2 and n == -2 then (
        (T,C) = randomK3surfaceContainingConic(g,CoefficientRing=>K);
        H := if g != 10 and g != 12 then Var ideal first gens ring ambient T else null;
        return latticePolarizedK3surface(T,C,H,{g,d,n});
    );
    if member(g,{3,4,5,6,7,8,9,10,12}) and d == 1 and n == -2 then ( 
        (X,C) = randomMukaiThreefoldContainingLine(g,CoefficientRing=>K);
        j = parametrize random({1},C);
        (T,C) = (j^* X,j^* C);
        return latticePolarizedK3surface(T,C,{g,d,n});
    );
    if (member(g,{3,4,5}) and d >= 3 and n == -2) and (g != 5 or d <= 8) and (g != 4 or d <= 6) and (g != 3 or d <= 8) then (
        C = randomRationalCurve(d,g,CoefficientRing=>K); 
        T = random(if g == 3 then {4} else if g == 4 then {{2},{3}} else {{2},{2},{2}},C);
        return latticePolarizedK3surface(T,C,{g,d,n});
    );
    if (member(g,{3,4,5}) and member(d,{3,4,5,6,7,8,9}) and n == 0) and (g != 5 or d <= 9) and (g != 4 or d <= 7) and (g != 3 or d <= 8) then (
        C = randomEllipticCurve(d,g,CoefficientRing=>K);
        T = random(if g == 3 then {4} else if g == 4 then {{2},{3}} else {{2},{2},{2}},C);
        return latticePolarizedK3surface(T,C,{g,d,n});
    );
    if member(g,{3,4,5,6,7,8,9,10,12}) and d == 0 and n == -2 then (
        (X,p) = randomPointedMukaiThreefold(g,CoefficientRing=>K);
        j = parametrize random({1},tangentSpace(X,p));
        T = j^* X; 
        C = j^* p; -- the node
        return latticePolarizedK3surface(T,C,{g,d,n});
    );
    if g >= 5 and d == 3 and n == 0 then return trigonalK3(g,CoefficientRing=>o.CoefficientRing);
    if g >= 3 and d == 4 and n == 0 then return tetragonalK3(g,CoefficientRing=>o.CoefficientRing);
    if g >= 3 and d == 5 and n == 0 then return pentagonalK3(g,CoefficientRing=>o.CoefficientRing);    
    error ("no procedure found to construct K3 surface with rank 2 lattice defined by the "|toString(matrix{{2*g-2,d},{d,n}}));
);

K3 EmbeddedK3surface := o -> T -> (
    if T.cache#?"AssociatedLatticePolarizedK3surface" then return T.cache#"AssociatedLatticePolarizedK3surface";
    (S,ab) := construction T;
    (a,b) := ab;
    f := map(S,a,b);    
    C := f(S#"curve");
    if not(dim C == 1 and isSubset(C,T)) then error "something went wrong on the curve image";
    if degree C != a * (S#"lattice")_(0,1) + b * (S#"lattice")_(1,1) then error "something went wrong on the degree of the curve image";
    T.cache#"AssociatedLatticePolarizedK3surface" = latticePolarizedK3surface(T,C,{genus T,degree C,(S#"lattice")_(1,1)})
);

trigonalK3 = method(TypicalValue => LatticePolarizedK3surface, Options => {CoefficientRing => ZZ/65521});
trigonalK3 ZZ := o -> g -> (
    -- See also [Beauville - A remark on the generalized Franchetta conjecture for K3 surfaces]
    if g < 5 then error "expected genus to be at least 5"; 
    n := (g-1)%3 + 1; if n == 1 then n = 4;
    a := lift((g-n)/3,ZZ);
    K := o.CoefficientRing;
    P := PP_K^{1,n};
    S := if n == 2 then random({2,3},0_P) else 
         if n == 3 then random({{1,1},{1,3}},0_P) else 
         if n == 4 then random({{0,3},{1,1},{1,1}},0_P);
    f := rationalMap((0_P)%S,{a,1});
    T := image f;  
    assert(dim ambient T == g and dim T == 2 and degree T == 2*g-2 and sectionalGenus T == g);
    pr1 := multirationalMap first projections S;
    C := f(pr1^* point target pr1);
    assert(dim C == 1 and degree C == 3);
    latticePolarizedK3surface(T,C,{g,3,0})
);

tetragCase1 = (K,a,g) -> (
    x := local x;
    R := K[x_0..x_5,Degrees=>{3:{1,0},1:{1,1},2:{0,1}}];
    U := ideal(random({2,1},R),random({2,2},R));
    M := basis({1,a},R);
    y := local y;
    T := K[y_0..y_(numColumns M -1)];
    j := map(quotient U,T,M);
    S := Var trim kernel j;
    assert(dim ambient S == g and dim S == 2 and degree S == 2*g-2 and sectionalGenus S == g);
    C := Var trim preimage(j,sub(ideal(x_4),target j));
    assert(dim C == 1 and degree C == 4);
    latticePolarizedK3surface(S,C,{g,4,0})
);

tetragonalK3 = method(TypicalValue => LatticePolarizedK3surface, Options => {CoefficientRing => ZZ/65521});
tetragonalK3 ZZ := o -> g -> (
    if g < 3 then error "expected genus to be at least 3";
    n := (g-1)%4;
    a := lift((g-1-n)/4,ZZ);
    K := o.CoefficientRing;
    if n == 0 then return K3 (K3(5,4,0))(1,a-1);
    if n == 1 then return tetragCase1(K,a,g);
    if n == 2 then return K3 (K3(3,4,0))(1,a);  
    if n == 3 then return K3 (K3(4,4,0))(1,a);
);

pentagCase0 = (K,a,g) -> (
    x := local x;
    y := local y;
    R := K[x_0..x_4,y_0,y_1,Degrees=>{2:{1,0},3:{1,1},2:{0,1}}];
    M := basis({1,a},R);
    z := local z;
    T := K[z_0..z_(numColumns M -1)];
    A := matrix pack(5,for i to 24 list random({1,1},R));
    A = A - transpose A;
    j := map(quotient pfaffians(4,A),T,M);
    S := Var kernel(j,2); -- trim kernel j; -- Warning: assume that S is cut out by quadrics 
    assert(dim ambient S == g and dim S == 2 and degree S == 2*g-2 and sectionalGenus S == g);
    C := Var trim preimage(j,sub(ideal(y_0),target j));
    assert(dim C == 1 and degree C == 5);
    latticePolarizedK3surface(S,C,{g,5,0})
)

pentagCase1 = (K,a,g) -> (
    x := local x;
    y := local y;
    R := K[x_0..x_4,y_0,y_1,Degrees=>{3:{1,0},2:{1,1},2:{0,1}}];
    M := basis({1,a},R);
    z := local z;
    T := K[z_0..z_(numColumns M -1)];
    A := matrix {{0,0,random({1,1},R),random({1,1},R),random({1,1},R)},
                 {0,0,random({1,1},R),random({1,1},R),random({1,1},R)},
                 {0,0,0              ,random({1,0},R),random({1,0},R)},
                 {0,0,0              ,0              ,random({1,0},R)},
                 {0,0,0              ,0              ,0              }};
    A = A - transpose(A);             
    j := map(quotient pfaffians(4,A),T,M);
    S := Var trim(kernel(j,2)); -- trim kernel j; -- Warning: assume that S is cut out by quadrics
    assert(dim ambient S == g and dim S == 2 and degree S == 2*g-2 and sectionalGenus S == g);
    C := Var trim preimage(j,sub(ideal(y_0),target j));
    assert(dim C == 1 and degree C == 5);
    latticePolarizedK3surface(S,C,{g,5,0})
);

pentagonalK3 = method(TypicalValue => LatticePolarizedK3surface, Options => {CoefficientRing => ZZ/65521});
pentagonalK3 ZZ := o -> g -> (
    if g < 3 then error "expected genus to be at least 3";
    n := (g-1)%5;
    a := lift((g-1-n)/5,ZZ);
    K := o.CoefficientRing;
    if n == 0 then return pentagCase0(K,a,g);
    if n == 1 then return pentagCase1(K,a,g);
    if n == 2 then return K3 (K3(3,5,0))(1,a);
    if n == 3 then return K3 (K3(4,5,0))(1,a);
    if n == 4 then return K3 (K3(5,5,0))(1,a);
);

--#######################################
-*
MAXg = 100;
MAXab = 300;
possible'a'b := (g,d,n) -> (
    local g'; local c;
    L := {};
    for a from 1 to MAXab do
        for b from 0 to MAXab do (
            g' = (a^2*(2*g-2) + 2*a*b*d + b^2*n + 2)/2;
            c = a*d + b*n;
            if floor g' == g' and g' >= g and c > 0 and g' <= MAXg then L = append(L,(floor g',c,a,b));
        );
    L      
);
possibilities = {};
for g from 3 to MAXg do for d from 1 to 15 do for n in {-2,0} do 
    if (g >= 3 and g <= 12 and d == 2 and n == -2) or
       (member(g,{3,4,5,6,7,8,9,10,12}) and d == 1 and n == -2) or 
       ((member(g,{3,4,5}) and d >= 3 and n == -2) and (g != 5 or d <= 8) and (g != 4 or d <= 6) and (g != 3 or d <= 8)) or
       ((member(g,{3,4,5}) and member(d,{3,4,5,6,7,8,9}) and n == 0) and (g != 5 or d <= 9) and (g != 4 or d <= 7) and (g != 3 or d <= 8)) or
       (member(g,{3,4,5,6,7,8,9,10,12}) and d == 0 and n == -2) or
       (g >= 5 and d == 3 and n == 0) or
       (g >= 3 and d == 4 and n == 0) or
       (g >= 3 and d == 5 and n == 0) 
then possibilities = append(possibilities,(g,d,n));
possibilities = sort flatten for l in possibilities list apply(possible'a'b l,u -> append(append(append(u,l_0),l_1),l_2));
-- possibilities = {...,(g',c,a,b,g,d,n),...}
*-
--#######################################
possibilities = {(3,1,1,0,3,1,-2), (3,2,1,0,3,2,-2), (3,3,1,0,3,3,-2), (3,3,1,0,3,3,0), (3,4,1,0,3,4,-2), (3,4,1,0,3,4,0), (3,5,1,0,3,5,-2), (3,5,1,0,3,5,0), (3,6,1,0,3,6,-2), (3,6,1,0,3,6,0), (3,7,1,0,3,7,-2), (3,7,1,0,3,7,0), (3,8,1,0,3,8,-2), (3,8,1,0,3,8,0), (4,1,1,0,4,1,-2), (4,2,1,0,4,2,-2), (4,3,1,0,4,3,-2), (4,3,1,0,4,3,0), (4,4,1,0,4,4,-2), (4,4,1,0,4,4,0), (4,5,1,0,4,5,-2), (4,5,1,0,4,5,0), (4,6,1,0,4,6,-2), (4,6,1,0,4,6,0), (4,7,1,0,4,7,0), (5,1,1,0,5,1,-2), (5,1,1,1,3,3,-2), (5,2,1,0,5,2,-2), (5,3,1,0,5,3,-2), (5,3,1,0,5,3,0), (5,4,1,0,5,4,-2), (5,4,1,0,5,4,0), (5,5,1,0,5,5,-2), (5,5,1,0,5,5,0), (5,6,1,0,5,6,-2), (5,6,1,0,5,6,0), (5,7,1,0,5,7,-2), (5,7,1,0,5,7,0), (5,8,1,0,5,8,-2), (5,8,1,0,5,8,0), (5,9,1,0,5,9,0), (6,1,1,0,6,1,-2), (6,1,1,1,4,3,-2), (6,2,1,0,6,2,-2), (6,2,1,1,3,4,-2), (6,3,1,0,6,3,0), (6,3,1,1,3,3,0), (6,4,1,0,6,4,0), (6,5,1,0,6,5,0), (7,1,1,0,7,1,-2), (7,1,1,1,5,3,-2), (7,2,1,0,7,2,-2), (7,2,1,1,4,4,-2), (7,3,1,0,7,3,0), (7,3,1,1,3,5,-2), (7,3,1,1,4,3,0), (7,4,1,0,7,4,0), (7,4,1,1,3,4,0), (7,5,1,0,7,5,0), (8,1,1,0,8,1,-2), (8,2,1,0,8,2,-2), (8,2,1,1,5,4,-2), (8,3,1,0,8,3,0), (8,3,1,1,4,5,-2), (8,3,1,1,5,3,0), (8,4,1,0,8,4,0), (8,4,1,1,3,6,-2), (8,4,1,1,4,4,0), (8,5,1,0,8,5,0), (8,5,1,1,3,5,0), (9,1,1,0,9,1,-2), (9,1,1,2,3,5,-2), (9,2,1,0,9,2,-2), (9,2,2,0,3,1,-2), (9,3,1,0,9,3,0), (9,3,1,1,5,5,-2), (9,3,1,1,6,3,0), (9,3,1,2,3,3,0), (9,4,1,0,9,4,0), (9,4,1,1,4,6,-2), (9,4,1,1,5,4,0), (9,4,2,0,3,2,-2), (9,5,1,0,9,5,0), (9,5,1,1,3,7,-2), (9,5,1,1,4,5,0), (9,6,1,1,3,6,0), (9,6,2,0,3,3,-2), (9,6,2,0,3,3,0), (9,8,2,0,3,4,-2), (9,8,2,0,3,4,0), (9,10,2,0,3,5,-2), (9,10,2,0,3,5,0), (9,12,2,0,3,6,-2), (9,12,2,0,3,6,0), (9,14,2,0,3,7,-2), (9,14,2,0,3,7,0), (9,16,2,0,3,8,-2), (9,16,2,0,3,8,0), (10,1,1,0,10,1,-2), (10,1,1,2,4,5,-2), (10,2,1,0,10,2,-2), (10,3,1,0,10,3,0), (10,3,1,1,7,3,0), (10,3,1,2,4,3,0), (10,4,1,0,10,4,0), (10,4,1,1,5,6,-2), (10,4,1,1,6,4,0), (10,5,1,0,10,5,0), (10,5,1,1,5,5,0), (10,6,1,1,3,8,-2), (10,6,1,1,4,6,0), (10,7,1,1,3,7,0), (11,1,1,2,5,5,-2), (11,2,1,0,11,2,-2), (11,2,1,2,3,6,-2), (11,3,1,0,11,3,0), (11,3,1,1,8,3,0), (11,3,1,2,5,3,0), (11,4,1,0,11,4,0), (11,4,1,1,7,4,0), (11,4,1,2,3,4,0), (11,5,1,0,11,5,0), (11,5,1,1,5,7,-2), (11,5,1,1,6,5,0), (11,6,1,1,5,6,0), (11,7,1,1,4,7,0), (11,8,1,1,3,8,0), (12,1,1,0,12,1,-2), (12,2,1,0,12,2,-2), (12,2,1,2,4,6,-2), (12,2,2,1,3,2,-2), (12,3,1,0,12,3,0), (12,3,1,1,9,3,0), (12,3,1,2,6,3,0), (12,3,1,3,3,3,0), (12,4,1,0,12,4,0), (12,4,1,1,8,4,0), (12,4,1,2,4,4,0), (12,5,1,0,12,5,0), (12,5,1,1,7,5,0), (12,6,1,1,5,8,-2), (12,7,1,1,5,7,0), (13,2,1,2,5,6,-2), (13,2,2,0,4,1,-2), (13,3,1,0,13,3,0), (13,3,1,1,10,3,0), (13,3,1,2,3,7,-2), (13,3,1,2,7,3,0), (13,3,1,3,4,3,0), (13,4,1,0,13,4,0), (13,4,1,1,9,4,0), (13,4,1,2,5,4,0), (13,4,2,0,4,2,-2), (13,5,1,0,13,5,0), (13,5,1,1,8,5,0), (13,5,1,2,3,5,0), (13,6,2,0,4,3,-2), (13,6,2,0,4,3,0), (13,8,1,1,5,8,0), (13,8,2,0,4,4,-2), (13,8,2,0,4,4,0), (13,10,2,0,4,5,-2), (13,10,2,0,4,5,0), (13,12,2,0,4,6,-2), (13,12,2,0,4,6,0), (13,14,2,0,4,7,0), (14,3,1,0,14,3,0), (14,3,1,1,11,3,0), (14,3,1,2,8,3,0), (14,3,1,3,5,3,0), (14,4,1,0,14,4,0), (14,4,1,1,10,4,0), (14,4,1,2,6,4,0), (14,4,2,1,3,3,-2), (14,5,1,0,14,5,0), (14,5,1,1,9,5,0), (14,5,1,2,4,5,0), (14,9,1,1,5,9,0), (15,1,1,3,3,7,-2), (15,3,1,0,15,3,0), (15,3,1,1,12,3,0), (15,3,1,2,5,7,-2), (15,3,1,2,9,3,0), (15,3,1,3,6,3,0), (15,3,1,4,3,3,0), (15,4,1,0,15,4,0), (15,4,1,1,11,4,0), (15,4,1,2,3,8,-2), (15,4,1,2,7,4,0), (15,4,1,3,3,4,0), (15,5,1,0,15,5,0), (15,5,1,1,10,5,0), (15,5,1,2,5,5,0), (15,6,1,2,3,6,0), (15,6,2,1,3,3,0), (16,2,2,1,4,2,-2), (16,3,1,0,16,3,0), (16,3,1,1,13,3,0), (16,3,1,2,10,3,0), (16,3,1,3,7,3,0), (16,3,1,4,4,3,0), (16,4,1,0,16,4,0), (16,4,1,1,12,4,0), (16,4,1,2,8,4,0), (16,4,1,3,4,4,0), (16,5,1,0,16,5,0), (16,5,1,1,11,5,0), (16,5,1,2,6,5,0), (16,6,1,2,4,6,0), (16,6,2,1,3,4,-2), (17,1,1,3,5,7,-2), (17,2,2,0,5,1,-2), (17,2,2,2,3,3,-2), (17,3,1,0,17,3,0), (17,3,1,1,14,3,0), (17,3,1,2,11,3,0), (17,3,1,3,8,3,0), (17,3,1,4,5,3,0), (17,4,1,0,17,4,0), (17,4,1,1,13,4,0), (17,4,1,2,5,8,-2), (17,4,1,2,9,4,0), (17,4,1,3,5,4,0), (17,4,2,0,5,2,-2), (17,5,1,0,17,5,0), (17,5,1,1,12,5,0), (17,5,1,2,7,5,0), (17,6,1,2,5,6,0), (17,6,2,0,5,3,-2), (17,6,2,0,5,3,0), (17,7,1,2,3,7,0), (17,8,2,0,5,4,-2), (17,8,2,0,5,4,0), (17,8,2,1,3,4,0), (17,10,2,0,5,5,-2), (17,10,2,0,5,5,0), (17,12,2,0,5,6,-2), (17,12,2,0,5,6,0), (17,14,2,0,5,7,-2), (17,14,2,0,5,7,0), (17,16,2,0,5,8,-2), (17,16,2,0,5,8,0), (17,18,2,0,5,9,0), (18,2,1,3,3,8,-2), (18,3,1,0,18,3,0), (18,3,1,1,15,3,0), (18,3,1,2,12,3,0), (18,3,1,3,9,3,0), (18,3,1,4,6,3,0), (18,3,1,5,3,3,0), (18,4,1,0,18,4,0), (18,4,1,1,14,4,0), (18,4,1,2,10,4,0), (18,4,1,3,6,4,0), (18,4,2,1,4,3,-2), (18,5,1,0,18,5,0), (18,5,1,1,13,5,0), (18,5,1,2,8,5,0), (18,5,1,3,3,5,0), (18,7,1,2,4,7,0), (18,8,2,1,3,5,-2), (19,3,1,0,19,3,0), (19,3,1,1,16,3,0), (19,3,1,2,13,3,0), (19,3,1,3,10,3,0), (19,3,1,4,7,3,0), (19,3,1,5,4,3,0), (19,3,3,0,3,1,-2), (19,4,1,0,19,4,0), (19,4,1,1,15,4,0), (19,4,1,2,11,4,0), (19,4,1,3,7,4,0), (19,4,1,4,3,4,0), (19,5,1,0,19,5,0), (19,5,1,1,14,5,0), (19,5,1,2,9,5,0), (19,5,1,3,4,5,0), (19,6,2,1,4,3,0), (19,6,3,0,3,2,-2), (19,7,1,2,5,7,0), (19,8,1,2,3,8,0), (19,9,3,0,3,3,-2), (19,9,3,0,3,3,0), (19,10,2,1,3,5,0), (19,12,3,0,3,4,-2), (19,12,3,0,3,4,0), (19,15,3,0,3,5,-2), (19,15,3,0,3,5,0), (19,18,3,0,3,6,-2), (19,18,3,0,3,6,0), (19,21,3,0,3,7,-2), (19,21,3,0,3,7,0), (19,24,3,0,3,8,-2), (19,24,3,0,3,8,0), (20,2,1,3,5,8,-2), (20,2,2,1,5,2,-2), (20,3,1,0,20,3,0), (20,3,1,1,17,3,0), (20,3,1,2,14,3,0), (20,3,1,3,11,3,0), (20,3,1,4,8,3,0), (20,3,1,5,5,3,0), (20,4,1,0,20,4,0), (20,4,1,1,16,4,0), (20,4,1,2,12,4,0), (20,4,1,3,8,4,0), (20,4,1,4,4,4,0), (20,5,1,0,20,5,0), (20,5,1,1,15,5,0), (20,5,1,2,10,5,0), (20,5,1,3,5,5,0), (20,6,2,1,4,4,-2), (20,10,2,1,3,6,-2), (21,1,3,1,3,1,-2), (21,2,2,0,6,1,-2), (21,2,2,2,4,3,-2), (21,3,1,0,21,3,0), (21,3,1,1,18,3,0), (21,3,1,2,15,3,0), (21,3,1,3,12,3,0), (21,3,1,4,9,3,0), (21,3,1,5,6,3,0), (21,3,1,6,3,3,0), (21,4,1,0,21,4,0), (21,4,1,1,17,4,0), (21,4,1,2,13,4,0), (21,4,1,3,9,4,0), (21,4,1,4,5,4,0), (21,4,2,0,6,2,-2), (21,4,2,2,3,4,-2), (21,5,1,0,21,5,0), (21,5,1,1,16,5,0), (21,5,1,2,11,5,0), (21,5,1,3,6,5,0), (21,6,1,3,3,6,0), (21,6,2,0,6,3,0), (21,6,2,2,3,3,0), (21,8,1,2,5,8,0), (21,8,2,0,6,4,0), (21,8,2,1,4,4,0), (21,10,2,0,6,5,0), (21,12,2,1,3,6,0), (22,3,1,0,22,3,0), (22,3,1,1,19,3,0), (22,3,1,2,16,3,0), (22,3,1,3,13,3,0), (22,3,1,4,10,3,0), (22,3,1,5,7,3,0), (22,3,1,6,4,3,0), (22,4,1,0,22,4,0), (22,4,1,1,18,4,0), (22,4,1,2,14,4,0), (22,4,1,3,10,4,0), (22,4,1,4,6,4,0), (22,4,2,1,5,3,-2), (22,5,1,0,22,5,0), (22,5,1,1,17,5,0), (22,5,1,2,12,5,0), (22,5,1,3,7,5,0), (22,6,1,3,4,6,0), (22,8,2,1,4,5,-2), (22,12,2,1,3,7,-2), (23,3,1,0,23,3,0), (23,3,1,1,20,3,0), (23,3,1,2,17,3,0), (23,3,1,3,14,3,0), (23,3,1,4,11,3,0), (23,3,1,5,8,3,0), (23,3,1,6,5,3,0), (23,4,1,0,23,4,0), (23,4,1,1,19,4,0), (23,4,1,2,15,4,0), (23,4,1,3,11,4,0), (23,4,1,4,7,4,0), (23,4,1,5,3,4,0), (23,5,1,0,23,5,0), (23,5,1,1,18,5,0), (23,5,1,2,13,5,0), (23,5,1,3,8,5,0), (23,5,1,4,3,5,0), (23,6,1,3,5,6,0), (23,6,2,1,5,3,0), (23,9,1,2,5,9,0), (23,10,2,1,4,5,0), (23,14,2,1,3,7,0), (24,2,2,1,6,2,-2), (24,2,2,3,3,4,-2), (24,3,1,0,24,3,0), (24,3,1,1,21,3,0), (24,3,1,2,18,3,0), (24,3,1,3,15,3,0), (24,3,1,4,12,3,0), (24,3,1,5,9,3,0), (24,3,1,6,6,3,0), (24,3,1,7,3,3,0), (24,4,1,0,24,4,0), (24,4,1,1,20,4,0), (24,4,1,2,16,4,0), (24,4,1,3,12,4,0), (24,4,1,4,8,4,0), (24,4,1,5,4,4,0), (24,4,3,1,3,2,-2), (24,5,1,0,24,5,0), (24,5,1,1,19,5,0), (24,5,1,2,14,5,0), (24,5,1,3,9,5,0), (24,5,1,4,4,5,0), (24,6,2,1,5,4,-2), (24,7,1,3,3,7,0), (24,10,2,1,4,6,-2), (24,14,2,1,3,8,-2), (25,2,2,0,7,1,-2), (25,2,2,2,5,3,-2), (25,3,1,0,25,3,0), (25,3,1,1,22,3,0), (25,3,1,2,19,3,0), (25,3,1,3,16,3,0), (25,3,1,4,13,3,0), (25,3,1,5,10,3,0), (25,3,1,6,7,3,0), (25,3,1,7,4,3,0), (25,4,1,0,25,4,0), (25,4,1,1,21,4,0), (25,4,1,2,17,4,0), (25,4,1,3,13,4,0), (25,4,1,4,9,4,0), (25,4,1,5,5,4,0), (25,4,2,0,7,2,-2), (25,4,2,2,4,4,-2), (25,5,1,0,25,5,0), (25,5,1,1,20,5,0), (25,5,1,2,15,5,0), (25,5,1,3,10,5,0), (25,5,1,4,5,5,0), (25,6,2,0,7,3,0), (25,6,2,2,3,5,-2), (25,6,2,2,4,3,0), (25,7,1,3,4,7,0), (25,8,2,0,7,4,0), (25,8,2,1,5,4,0), (25,8,2,2,3,4,0), (25,10,2,0,7,5,0), (25,12,2,1,4,6,0), (25,16,2,1,3,8,0), (26,3,1,0,26,3,0), (26,3,1,1,23,3,0), (26,3,1,2,20,3,0), (26,3,1,3,17,3,0), (26,3,1,4,14,3,0), (26,3,1,5,11,3,0), (26,3,1,6,8,3,0), (26,3,1,7,5,3,0), (26,4,1,0,26,4,0), (26,4,1,1,22,4,0), (26,4,1,2,18,4,0), (26,4,1,3,14,4,0), (26,4,1,4,10,4,0), (26,4,1,5,6,4,0), (26,5,1,0,26,5,0), (26,5,1,1,21,5,0), (26,5,1,2,16,5,0), (26,5,1,3,11,5,0), (26,5,1,4,6,5,0), (26,7,1,3,5,7,0), (26,8,2,1,5,5,-2), (27,2,3,2,3,2,-2), (27,3,1,0,27,3,0), (27,3,1,1,24,3,0), (27,3,1,2,21,3,0), (27,3,1,3,18,3,0), (27,3,1,4,15,3,0), (27,3,1,5,12,3,0), (27,3,1,6,9,3,0), (27,3,1,7,6,3,0), (27,3,1,8,3,3,0), (27,4,1,0,27,4,0), (27,4,1,1,23,4,0), (27,4,1,2,19,4,0), (27,4,1,3,15,4,0), (27,4,1,4,11,4,0), (27,4,1,5,7,4,0), (27,4,1,6,3,4,0), (27,5,1,0,27,5,0), (27,5,1,1,22,5,0), (27,5,1,2,17,5,0), (27,5,1,3,12,5,0), (27,5,1,4,7,5,0), (27,6,1,4,3,6,0), (27,6,2,1,6,3,0), (27,6,2,3,3,3,0), (27,7,3,1,3,3,-2), (27,8,1,3,3,8,0), (27,10,2,1,5,5,0), (27,14,2,1,4,7,0), (28,2,2,1,7,2,-2), (28,2,2,3,4,4,-2), (28,3,1,0,28,3,0), (28,3,1,1,25,3,0), (28,3,1,2,22,3,0), (28,3,1,3,19,3,0), (28,3,1,4,16,3,0), (28,3,1,5,13,3,0), (28,3,1,6,10,3,0), (28,3,1,7,7,3,0), (28,3,1,8,4,3,0), (28,3,3,0,4,1,-2), (28,4,1,0,28,4,0), (28,4,1,1,24,4,0), (28,4,1,2,20,4,0), (28,4,1,3,16,4,0), (28,4,1,4,12,4,0), (28,4,1,5,8,4,0), (28,4,1,6,4,4,0), (28,5,1,0,28,5,0), (28,5,1,1,23,5,0), (28,5,1,2,18,5,0), (28,5,1,3,13,5,0), (28,5,1,4,8,5,0), (28,5,1,5,3,5,0), (28,6,1,4,4,6,0), (28,6,3,0,4,2,-2), (28,9,3,0,4,3,-2), (28,9,3,0,4,3,0), (28,9,3,1,3,3,0), (28,10,2,1,5,6,-2), (28,12,3,0,4,4,-2), (28,12,3,0,4,4,0), (28,15,3,0,4,5,-2), (28,15,3,0,4,5,0), (28,18,3,0,4,6,-2), (28,18,3,0,4,6,0), (28,21,3,0,4,7,0), (29,2,2,0,8,1,-2), (29,3,1,0,29,3,0), (29,3,1,1,26,3,0), 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(97,10,2,2,20,5,0), (97,10,2,4,15,5,0), (97,10,2,6,10,5,0), (97,10,2,8,5,5,0), (97,12,2,7,4,6,0), (97,12,3,2,9,4,0), (97,12,3,5,5,4,0), (97,12,4,0,7,3,0), (97,12,4,4,3,5,-2), (97,12,4,4,4,3,0), (97,14,2,6,4,7,0), (97,15,3,1,10,5,0), (97,15,3,4,5,5,0), (97,16,2,5,5,8,0), (97,16,4,0,7,4,0), (97,16,4,1,6,4,0), (97,16,4,2,5,4,0), (97,16,4,3,4,4,0), (97,16,4,4,3,4,0), (97,20,4,0,7,5,0), (97,21,5,2,3,5,-2), (97,24,4,2,4,6,0), (97,24,6,1,3,4,0), (97,32,4,1,5,8,0), (97,32,4,2,3,8,0), (98,3,1,0,98,3,0), (98,3,1,1,95,3,0), (98,3,1,2,92,3,0), (98,3,1,3,89,3,0), (98,3,1,4,86,3,0), (98,3,1,5,83,3,0), (98,3,1,6,80,3,0), (98,3,1,7,77,3,0), (98,3,1,8,74,3,0), (98,3,1,9,71,3,0), (98,3,1,10,68,3,0), (98,3,1,11,65,3,0), (98,3,1,12,62,3,0), (98,3,1,13,59,3,0), (98,3,1,14,56,3,0), (98,3,1,15,53,3,0), (98,3,1,16,50,3,0), (98,3,1,17,47,3,0), (98,3,1,18,44,3,0), (98,3,1,19,41,3,0), (98,3,1,20,38,3,0), (98,3,1,21,35,3,0), (98,3,1,22,32,3,0), (98,3,1,23,29,3,0), (98,3,1,24,26,3,0), (98,3,1,25,23,3,0), (98,3,1,26,20,3,0), (98,3,1,27,17,3,0), (98,3,1,28,14,3,0), (98,3,1,29,11,3,0), (98,3,1,30,8,3,0), (98,3,1,31,5,3,0), (98,4,1,0,98,4,0), (98,4,1,1,94,4,0), (98,4,1,2,90,4,0), (98,4,1,3,86,4,0), (98,4,1,4,82,4,0), (98,4,1,5,78,4,0), (98,4,1,6,74,4,0), (98,4,1,7,70,4,0), (98,4,1,8,66,4,0), (98,4,1,9,62,4,0), (98,4,1,10,58,4,0), (98,4,1,11,54,4,0), (98,4,1,12,50,4,0), (98,4,1,13,46,4,0), (98,4,1,14,42,4,0), (98,4,1,15,38,4,0), (98,4,1,16,34,4,0), (98,4,1,17,30,4,0), (98,4,1,18,26,4,0), (98,4,1,19,22,4,0), (98,4,1,20,18,4,0), (98,4,1,21,14,4,0), (98,4,1,22,10,4,0), (98,4,1,23,6,4,0), (98,5,1,0,98,5,0), (98,5,1,1,93,5,0), (98,5,1,2,88,5,0), (98,5,1,3,83,5,0), (98,5,1,4,78,5,0), (98,5,1,5,73,5,0), (98,5,1,6,68,5,0), (98,5,1,7,63,5,0), (98,5,1,8,58,5,0), (98,5,1,9,53,5,0), (98,5,1,10,48,5,0), (98,5,1,11,43,5,0), (98,5,1,12,38,5,0), (98,5,1,13,33,5,0), (98,5,1,14,28,5,0), (98,5,1,15,23,5,0), (98,5,1,16,18,5,0), (98,5,1,17,13,5,0), (98,5,1,18,8,5,0), (98,5,1,19,3,5,0), (99,2,3,2,11,2,-2), (99,2,3,8,3,6,-2), (99,3,1,0,99,3,0), (99,3,1,1,96,3,0), (99,3,1,2,93,3,0), (99,3,1,3,90,3,0), (99,3,1,4,87,3,0), (99,3,1,5,84,3,0), (99,3,1,6,81,3,0), (99,3,1,7,78,3,0), (99,3,1,8,75,3,0), (99,3,1,9,72,3,0), (99,3,1,10,69,3,0), (99,3,1,11,66,3,0), (99,3,1,12,63,3,0), (99,3,1,13,60,3,0), (99,3,1,14,57,3,0), (99,3,1,15,54,3,0), (99,3,1,16,51,3,0), (99,3,1,17,48,3,0), (99,3,1,18,45,3,0), (99,3,1,19,42,3,0), (99,3,1,20,39,3,0), (99,3,1,21,36,3,0), (99,3,1,22,33,3,0), (99,3,1,23,30,3,0), (99,3,1,24,27,3,0), (99,3,1,25,24,3,0), (99,3,1,26,21,3,0), (99,3,1,27,18,3,0), (99,3,1,28,15,3,0), (99,3,1,29,12,3,0), (99,3,1,30,9,3,0), (99,3,1,31,6,3,0), (99,3,1,32,3,3,0), (99,4,1,0,99,4,0), (99,4,1,1,95,4,0), (99,4,1,2,91,4,0), (99,4,1,3,87,4,0), (99,4,1,4,83,4,0), (99,4,1,5,79,4,0), (99,4,1,6,75,4,0), (99,4,1,7,71,4,0), (99,4,1,8,67,4,0), (99,4,1,9,63,4,0), (99,4,1,10,59,4,0), (99,4,1,11,55,4,0), (99,4,1,12,51,4,0), (99,4,1,13,47,4,0), (99,4,1,14,43,4,0), (99,4,1,15,39,4,0), (99,4,1,16,35,4,0), (99,4,1,17,31,4,0), (99,4,1,18,27,4,0), (99,4,1,19,23,4,0), (99,4,1,20,19,4,0), (99,4,1,21,15,4,0), (99,4,1,22,11,4,0), (99,4,1,23,7,4,0), (99,4,1,24,3,4,0), (99,5,1,0,99,5,0), (99,5,1,1,94,5,0), (99,5,1,2,89,5,0), (99,5,1,3,84,5,0), (99,5,1,4,79,5,0), (99,5,1,5,74,5,0), (99,5,1,6,69,5,0), (99,5,1,7,64,5,0), (99,5,1,8,59,5,0), (99,5,1,9,54,5,0), (99,5,1,10,49,5,0), (99,5,1,11,44,5,0), (99,5,1,12,39,5,0), (99,5,1,13,34,5,0), (99,5,1,14,29,5,0), (99,5,1,15,24,5,0), (99,5,1,16,19,5,0), (99,5,1,17,14,5,0), (99,5,1,18,9,5,0), (99,5,1,19,4,5,0), (99,6,1,16,3,6,0), (99,6,2,1,24,3,0), (99,6,2,3,21,3,0), (99,6,2,5,18,3,0), (99,6,2,7,15,3,0), (99,6,2,9,12,3,0), (99,6,2,11,9,3,0), (99,6,2,13,6,3,0), (99,6,2,15,3,3,0), (99,7,7,0,3,1,-2), (99,8,1,12,3,8,0), (99,10,2,1,23,5,0), (99,10,2,3,18,5,0), (99,10,2,5,13,5,0), (99,10,2,7,8,5,0), (99,10,2,9,3,5,0), (99,11,3,5,3,7,-2), (99,14,7,0,3,2,-2), (99,16,3,4,3,8,-2), (99,21,7,0,3,3,-2), (99,21,7,0,3,3,0), (99,28,7,0,3,4,-2), (99,28,7,0,3,4,0), (99,35,7,0,3,5,-2), (99,35,7,0,3,5,0), (99,42,7,0,3,6,-2), (99,42,7,0,3,6,0), (99,49,7,0,3,7,-2), (99,49,7,0,3,7,0), (99,56,7,0,3,8,-2), (99,56,7,0,3,8,0), (100,2,4,1,7,1,-2), (100,2,4,5,5,3,-2), (100,2,5,4,4,2,-2), (100,3,1,0,100,3,0), (100,3,1,1,97,3,0), (100,3,1,2,94,3,0), (100,3,1,3,91,3,0), (100,3,1,4,88,3,0), (100,3,1,5,85,3,0), (100,3,1,6,82,3,0), (100,3,1,7,79,3,0), (100,3,1,8,76,3,0), (100,3,1,9,73,3,0), (100,3,1,10,70,3,0), (100,3,1,11,67,3,0), (100,3,1,12,64,3,0), (100,3,1,13,61,3,0), (100,3,1,14,58,3,0), (100,3,1,15,55,3,0), (100,3,1,16,52,3,0), (100,3,1,17,49,3,0), (100,3,1,18,46,3,0), (100,3,1,19,43,3,0), (100,3,1,20,40,3,0), (100,3,1,21,37,3,0), (100,3,1,22,34,3,0), (100,3,1,23,31,3,0), (100,3,1,24,28,3,0), (100,3,1,25,25,3,0), (100,3,1,26,22,3,0), (100,3,1,27,19,3,0), (100,3,1,28,16,3,0), (100,3,1,29,13,3,0), (100,3,1,30,10,3,0), (100,3,1,31,7,3,0), (100,3,1,32,4,3,0), (100,3,3,0,12,1,-2), (100,4,1,0,100,4,0), (100,4,1,1,96,4,0), (100,4,1,2,92,4,0), (100,4,1,3,88,4,0), (100,4,1,4,84,4,0), (100,4,1,5,80,4,0), (100,4,1,6,76,4,0), (100,4,1,7,72,4,0), (100,4,1,8,68,4,0), (100,4,1,9,64,4,0), (100,4,1,10,60,4,0), (100,4,1,11,56,4,0), (100,4,1,12,52,4,0), (100,4,1,13,48,4,0), (100,4,1,14,44,4,0), (100,4,1,15,40,4,0), (100,4,1,16,36,4,0), (100,4,1,17,32,4,0), (100,4,1,18,28,4,0), (100,4,1,19,24,4,0), (100,4,1,20,20,4,0), (100,4,1,21,16,4,0), (100,4,1,22,12,4,0), (100,4,1,23,8,4,0), (100,4,1,24,4,4,0), (100,5,1,0,100,5,0), (100,5,1,1,95,5,0), (100,5,1,2,90,5,0), (100,5,1,3,85,5,0), (100,5,1,4,80,5,0), (100,5,1,5,75,5,0), (100,5,1,6,70,5,0), (100,5,1,7,65,5,0), (100,5,1,8,60,5,0), (100,5,1,9,55,5,0), (100,5,1,10,50,5,0), (100,5,1,11,45,5,0), (100,5,1,12,40,5,0), (100,5,1,13,35,5,0), (100,5,1,14,30,5,0), (100,5,1,15,25,5,0), (100,5,1,16,20,5,0), (100,5,1,17,15,5,0), (100,5,1,18,10,5,0), (100,5,1,19,5,5,0), (100,6,1,16,4,6,0), (100,6,3,0,12,2,-2), (100,6,3,6,4,6,-2), (100,6,6,3,3,2,-2), (100,9,3,0,12,3,0), (100,9,3,1,11,3,0), (100,9,3,2,10,3,0), (100,9,3,3,9,3,0), (100,9,3,4,8,3,0), (100,9,3,5,7,3,0), (100,9,3,6,6,3,0), (100,9,3,7,5,3,0), (100,9,3,8,4,3,0), (100,9,3,9,3,3,0), (100,12,3,0,12,4,0), (100,12,3,3,8,4,0), (100,12,3,6,4,4,0), (100,14,4,3,4,5,-2), (100,15,3,0,12,5,0), (100,15,3,3,7,5,0), (100,18,3,3,5,8,-2), (100,18,3,4,4,6,0), (100,21,3,3,5,7,0), (100,23,5,1,4,5,-2)};
assert(# possibilities == 5292);
fewPossibilities = select(possibilities,l -> first l <= 44);
assert(# fewPossibilities == 1157);

K3 String := o -> strG -> (
    G := value strG;
    if not instance(G,ZZ) then error "expected the string of an integer";
    if G > 100 then error "not implemented yet: show available functions to construct K3 surfaces of genus > 100";
    P := select(possibilities,l -> first l == G);
    if #P == 0 then error "no procedure found";
    if o.Verbose then (
        local g'; local c; local a; local b; local g; local d; local n;
        for s in P do (
            (g',c,a,b,g,d,n) = s;
            <<"(K3("<<g<<","<<d<<","<<n<<"))("<<a<<","<<b<<") -- K3 surface of genus "<<g'<<" and degree "<<2*g'-2<<" containing "<<(if n == -2 then "rational" else if n == 0 then "elliptic" else "")<<" curve of degree "<<c<<endl;
        );
    );
    return apply(P,l -> take(l,{4,6}));
);

K3 (String,VisibleList) := o -> (strG,opt) -> (
    if not #opt == 1 then error "option not available";
    opt = first opt;
    if first toList opt =!= Unique then error "Unique is the only available option for K3(String)";
    if not last opt then return K3 strG;    
    G := value strG;
    if not instance(G,ZZ) then error "expected the string of an integer";
    if G > 100 then error "not implemented yet: show available functions to construct K3 surfaces of genus > 100";
    P := select(possibilities,l -> first l == G);
    local g'; local c; local a; local b; local g; local d; local n;
    E := {};
    for s in P do (
        (g',c,a,b,g,d,n) = s;
        if not member((g',c,n),E) then (
            if not isprimitive(matrix{{2*g-2,d},{d,n}},a,b) then (
                -- <<"case (g',c,a,b,g,d,n) = "<<(g',c,a,b,g,d,n)<<" excluded as it does not give a primitive lattice embedding"<<endl;
                continue;
            );
            E = append(E,(g',c,n));
            if o.Verbose then <<"(K3("<<g<<","<<d<<","<<n<<"))("<<a<<","<<b<<") -- K3 surface of genus "<<g'<<" and degree "<<2*g'-2<<" containing "<<(if n == -2 then "rational" else if n == 0 then "elliptic" else "")<<" curve of degree "<<c<<endl;
        );
    );
    return E
);

randomPointedMukaiThreefold = method(Options => {CoefficientRing => ZZ/65521});
randomPointedMukaiThreefold ZZ := o -> g -> (
    local p; local X; local j; local psi;
    K := o.CoefficientRing;
    if g == 3 then (
        p = point PP_K^4;
        X = random({4},p);
        return (X,p);
    );
    if g == 4 then (
        p = point PP_K^5;
        X = random({{2},{3}},p);
        return (X,p);
    );
    if g == 5 then (
        p = point PP_K^6;
        X = random({{2},{2},{2}},p);
        return (X,p);
    );
    if g == 6 then (
        G14 := GG(K,1,4);
        p = schubertCycle((3,3),G14);
        j = parametrize random({{1},{1}},p);
        p = j^* p;
        X = (j^* G14) * random(2,p);
        return (X,p);
    );
    if g == 7 then (
        quinticDelPezzoSurface := Var image rationalMap(ring PP_K^2,{3,4});
        psi = rationalMap(quinticDelPezzoSurface * random(1,0_quinticDelPezzoSurface),2);
        p = psi point source psi;
        j = parametrize random({{1},{1}},p);
        X = j^* image(2,psi);
        p = j^* p;
        return (X,p);
    );
    if g == 8 then (
        G15 := GG(K,1,5);
        p = schubertCycle((4,4),G15);
        j = parametrize random({{1},{1},{1},{1},{1}},p);
        X = j^* G15;
        p = j^* p;
        return (X,p);
    );
    if g == 9 then (
        psi = rationalMap(image(PP_K^(2,2) << PP_K^6),3,2);
        p = psi point source psi;
        j = parametrize random({{1},{1},{1}},p);
        X = j^* image psi;
        p = j^* p;
        return (X,p);
    );
    if g == 10 then (
        XpLC10 := pointLineAndConicOnMukaiThreefoldOfGenus10(K,false,false);
        return (XpLC10_0,XpLC10_1);
    );
    if g == 12 then (
        XpLC12 := pointLineAndConicOnMukaiThreefoldOfGenus12(K,false,false);
        return (XpLC12_0,XpLC12_1);
    );
    error "the genus has to belong to the set {3,...,10,12}";
);

randomMukaiThreefoldContainingLine = method(Options => {CoefficientRing => ZZ/65521});
randomMukaiThreefoldContainingLine ZZ := o -> g -> (
    local L; local X; local j; local psi;
    K := o.CoefficientRing;
    if g == 3 then (
        L = random({{1},{1},{1}},0_(PP_K^4));
        X = random({4},L);
        return (X,L);
    );
    if g == 4 then (
        L = random({{1},{1},{1},{1}},0_(PP_K^5));
        X = random({{2},{3}},L);
        return (X,L);
    );
    if g == 5 then (
        L = random({{1},{1},{1},{1},{1}},0_(PP_K^6));
        X = random({{2},{2},{2}},L);
        return (X,L);
    );
    if g == 6 then (
        G14 := GG(K,1,4);
        L = schubertCycle((3,2),G14);
        j = parametrize random({{1},{1}},L);
        L = j^* L;
        X = (j^* G14) * random(2,L);
        return (X,L);
    );
    if g == 7 then (
        mapDP5 := multirationalMap rationalMap(ring PP_K^2,{3,4});
        pt := mapDP5 point source mapDP5;
        psi = rationalMap((image mapDP5) * random(1,pt),2);
        L = psi linearSpan {pt,point source psi};
        j = parametrize random({{1},{1}},L);
        X = j^* image(2,psi);
        L = j^* L;
        return (X,L);
    );
    if g == 8 then (
        G15 := GG(K,1,5);
        L = schubertCycle((4,3),G15);
        j = parametrize random({{1},{1},{1},{1},{1}},L);
        X = j^* G15;
        L = j^* L;
        return (X,L);
    );
    if g == 9 then (
        mapVerInP6 := (parametrize PP_K^(2,2)) << PP_K^6;
        psi = rationalMap(image mapVerInP6,3,2);
        L = psi linearSpan {mapVerInP6 point source mapVerInP6,point source psi};
        j = parametrize random({{1},{1},{1}},L);
        X = j^* image psi;
        L = j^* L;
        return (X,L);
    );
    if g == 10 then (
        XpLC10 := pointLineAndConicOnMukaiThreefoldOfGenus10(K,true,false);
        return (XpLC10_0,XpLC10_2);
    );
    if g == 12 then (
        XpLC12 := pointLineAndConicOnMukaiThreefoldOfGenus12(K,true,false);
        return (XpLC12_0,XpLC12_2);
    );
    error "the genus has to belong to the set {3,...,10,12}";
);

randomK3surfaceContainingConic = method(Options => {CoefficientRing => ZZ/65521});
randomK3surfaceContainingConic ZZ := o -> g -> (
    if not (g >= 3 and g <= 12) then error "expected integer between 3 and 12";
    local X; local T; local C; local p; local j;
    K := o.CoefficientRing;    
    if member(g,{3,4,5,6,7,8,9,11}) then (
        (X,p) = randomPointedMukaiThreefold(g+1,CoefficientRing=>K);
        j = parametrize random({1},tangentSpace(X,p));
        T = j^* X;
        p = (j^* p) % T;
        pr := rationalMap p; 
        (pr1,pr2) := graph pr;
        C = pr2 pr1^* p;
        T = image pr;
    );
    if g == 10 then (
        XpLC10 := pointLineAndConicOnMukaiThreefoldOfGenus10(K,false,true);
        (X,C) = (XpLC10_0,XpLC10_3);
        j = parametrize random({1},C);
        (T,C) = (j^* X,j^* C);
    );
    if g == 12 then (
        XpLC12 := pointLineAndConicOnMukaiThreefoldOfGenus12(K,false,true);
        (X,C) = (XpLC12_0,XpLC12_3);
        j = parametrize random({1},C);
        (T,C) = (j^* X,j^* C);
    );
    if not (dim C == 1 and degree C == 2 and sectionalGenus C == 0 and dim T == 2 and degree T == 2*g-2 and sectionalGenus T == g and isSubset(C,T)) then error "something went wrong";
    (T,C)
);

pointLineAndConicOnMukaiThreefoldOfGenus10 = (K,withLine,withConic) -> (
    line := null; conic := null;
    pts := apply(4,i -> point PP_K^2); 
    p := point PP_K^2;
    f := rationalMap((⋃ pts) % random(5,p + ⋃ apply(pts,i -> 2*i)),3);
    f = f * rationalMap point target f;
    -- C is a curve of degree 7 and genus 2 in P^4 and p is one of its points 
    C := image f;
    p = f p;
    if not(dim C == 1 and degree C == 7 and sectionalGenus C == 2 and dim p == 0 and degree p == 1 and isSubset(p,C)) then error "something went wrong";
    -- Q is a quadric in P^4 containing C and a point q
    q := point target f;
    Q := random(2,C + q);
    -- psi is the map Q --> PP^11 defined by the quintic hypersurfaces with double points along C
    psi := rationalMap(C_Q,5,2);
    if dim target psi != 11 then error "something went wrong";
    psi = toRationalMap psi;
    if K === ZZ/(char K) then interpolateImage(psi,toList(28:2),2) else forceImage(psi,image(2,psi));
    psi = multirationalMap psi;
    X := psi#"image";
    assert(X =!= null);
    q = psi q;
    if ? ideal q != "one-point scheme in PP^11" then error "something went wrong";
    if withConic then (
        -- the base locus of psi is supported on C and a reducible curve D, the union of 4 lines
        D := (Var trim lift(ideal matrix psi,ring ambient source psi)) \\ C;
        -- L is a line passing through p\in C which is contained in Q and intersects D in one point (thus psi L is a conic)
        -- (this needs to find a K-rational point on a certain 0-dimensional set defined over K)
        p' := select(decompose(D * coneOfLines(Q,p)),i -> dim i == 0 and degree i == 1);
        if # p' == 0 then error ("failed to find conic on Mukai threefold of genus 10 defined over "|toString(K));
        L := linearSpan {p,first p'};
        F := psi L;
        if not(dim F == 1 and degree F == 2 and flatten degrees ideal F == append(toList(9:1),2) and isSubset(F,X)) then error "something went wrong when trying to find conic on Mukai threefold of genus 10";
        conic = F;
    );
    if withLine then (
        psi = toRationalMap rationalMap(psi,Dominant=>true);
        -- B is the base locus of psi^-1, B-Supp(B) is a line.
        B := trim lift(ideal matrix approximateInverseMap(psi,Verbose=>false),ambient target psi);
        j := parametrize ideal matrix rationalMap(B,1);
        pr := rationalMap for i to 3 list random(1,source j);
        B' := pr j^* B;
        E' := B' : radical B';
        E := j radical trim (j^* ideal X + pr^* E');
        if not(dim E -1 == 1 and degree E == 1 and flatten degrees E == toList(10:1) and isSubset(ideal X,E)) then error "something went wrong when trying to find line on Mukai threefold of genus 10";
        line = Var E;
    );
    (X,q,line,conic)
);

pointLineAndConicOnMukaiThreefoldOfGenus12 = (K,withLine,withConic) -> (
    line := null; conic := null;
    f := rationalMap veronese(1,6,K);
    f = f * rationalMap(target f,ring PP_K^4,for i to 4 list random(1,target f));
    p := ideal point PP_K^4;    
    C := trim sub(image f,quotient ideal random(2,Var intersect(p,image f)));
    psi := rationalMap(saturate(C^2),5);
    if numgens target psi != 14 then error "something went wrong";
    if K === ZZ/(char K) then interpolateImage(psi,toList(45:2),2) else forceImage(psi,image(2,psi));
    X := psi#"idealImage";
    assert(X =!= null);
    p = psi p;
    if ? p != "one-point scheme in PP^13" then error "something went wrong";
    if withConic then (
        psi = rationalMap(psi,Dominant=>true);
        -- B is the base locus of psi^-1, B-Supp(B) is a conic.
        B := trim lift(ideal matrix approximateInverseMap(psi,Verbose=>false),ambient target psi);
        j := parametrize ideal matrix rationalMap(B,1);
        pr := rationalMap for i to 3 list random(1,source j);
        B' := pr j^* B;
        E' := B' : radical B';
        E := j radical trim (j^* X + pr^* E');
        if not(dim E -1 == 1 and degree E == 2 and flatten degrees E == {1,1,1,1,1,1,1,1,1,1,1,2} and isSubset(X,E)) then error "something went wrong when trying to find conic on Mukai threefold of genus 12";
        conic = E;
    );
    if withLine then (
        C = trim lift(C,ring PP_K^4);
        q := f point source f;
        -- L is a secant line to the sextic curve C contained in the quadric, source of psi
        q' := select(decompose saturate(C + ideal coneOfLines(Var ideal source psi,Var q),q),l -> dim l == 1 and degree l == 1);
        if # q' == 0 then error ("failed to find line on random Mukai threefold of genus 12 defined over "|toString(K));
        L := ideal image basis(1,intersect(q,first q'));
        F := psi L;
        if not(dim F -1 == 1 and degree F == 1 and flatten degrees F == toList(12:1) and isSubset(X,F)) then error "something went wrong when trying to find line on Mukai threefold of genus 12";
        line = F;
    );
    (Var X,Var p,if line === null then line else Var line,if conic === null then conic else Var conic)
);

power0 := method();
power0 (Ideal,ZZ) := (p,d) -> (
   assert(isPolynomialRing ambient ring p and isHomogeneous ideal ring p and isHomogeneous p and unique degrees p == {{1}});
   if d <= 0 then error "expected a positive integer";
   L := trim sub(ideal random(1,ambient ring p),ring p);
   if d == 1 or d == 2 or d == 3 then return saturate(p^d,L);
   if d == 4 then return saturate((power0(p,2))^2,L);
   if d == 5 then return saturate(power0(p,2) * power0(p,3),L);
   if d == 6 then return saturate(power0(p,3) * power0(p,3),L);
   if d == 7 then return saturate(saturate(power0(p,3) * power0(p,2),L) * power0(p,2),L);
   if d == 8 then return saturate(power0(p,4) * power0(p,4),L);
   if d > 8 then error "not implemented yet";
);

projectionInt = method();
projectionInt (VisibleList,RationalMap,ZZ,ZZ) := (L,phi,D,g) -> (
   try assert(ring matrix{L} === ZZ and #L>0) else error "expected a list of integers";
   E := invProjection(L,D,g,g);
   d := max flatten degrees ideal matrix phi;
   J := rationalMap(intersect apply(L,i -> power0(point source phi,i)),d);   
   f := rationalMap(intersect(ideal matrix phi,ideal matrix J),d);
   X := if char coefficientRing phi <= 65521 then Var image(f,"F4") else Var image f;
   <<endl;
   if E != (degree X,sectionalGenus X,dim ambient X) then <<"--warning: output is not as expected"<<endl else <<"-- (degree and genus are as expected)"<<endl;
   X
);

project = method();
project (VisibleList,LatticePolarizedK3surface,ZZ,ZZ) := (L,S,a,b) -> projectionInt(L,toRationalMap map(S,a,b),degree(S,a,b),genus(S,a,b));
project (VisibleList,EmbeddedK3surface) := (L,S) -> (
    if S.cache#?"GeneralK3" and S.cache#"GeneralK3" then return projectionInt(L,toRationalMap map S,degree S,genus S);
    (T,ab) := construction S;
    project(L,T,ab_0,ab_1)
);

invProjection = method();
invProjection (VisibleList,ZZ,ZZ,ZZ) := (mm,d,g,N) -> (
   expNumHyp := (d0,g0,r0,J) -> (
       t := local t;
       R := QQ[t];
       hP := (d,g) -> (1/2)*d*t^2 + ((1/2)*d+1-g)*t+2;
       max(binomial(r0+J,J) - sub(hP(d0,g0),t=>J),0)
   );
   mm = deepSplice mm; 
   <<"-- *** simulation ***"<<endl;
   <<"-- surface of degree "<<d<<" and sectional genus "<<g<<" in PP^"<<N<<" (quadrics: "<<expNumHyp(d,g,N,2)<<", cubics: "<<expNumHyp(d,g,N,3)<<")"<<endl;
   for m in mm do (
       (d,g,N) = (d - m^2,g - binomial(m,2),N - binomial(m+1,2)); 
       <<"-- surface of degree "<<d<<" and sectional genus "<<g<<" in PP^"<<N<<" (quadrics: "<<expNumHyp(d,g,N,2)<<", cubics: "<<expNumHyp(d,g,N,3)<<")"<<endl;
   );
   (d,g,N)
);

randomEllipticNormalCurve = method(Options => {CoefficientRing => ZZ/65521});
randomEllipticNormalCurve ZZ := o -> deg -> (
    K := o.CoefficientRing;
    S := ring PP_K^2;
    y := gens S;
    E := y_2^2*y_0-y_1^3+random(0,S)*y_1*y_0^2+random(0,S)*y_0^3;
    n := deg -1;
    nO := saturate((ideal(y_0,y_1))^(n+1)+ideal E);
    d := max flatten degrees nO;
    Hs := gens nO;    
    H := Hs*random(source Hs,S^{-d});
    linsys1 := gens truncate(d,(ideal H +ideal E):nO);
    linsys := mingens ideal (linsys1 % ideal E);
    SE := S/E;          
    phi := map(SE,ring PP_K^n,sub(linsys,SE));
    Var trim kernel phi
);

randomEllipticCurve = method(Options => {CoefficientRing => ZZ/65521});
randomEllipticCurve (ZZ,ZZ) := o -> (d,n) -> (
    K := o.CoefficientRing;
    C := randomEllipticNormalCurve(d,CoefficientRing=>K);
    f := rationalMap(ring PP_K^(d-1),ring PP_K^n,for i to n list random(1,ring PP_K^(d-1)));
    f C
);

randomRationalCurve = method(Options => {CoefficientRing => ZZ/65521});
randomRationalCurve (ZZ,ZZ) := o -> (d,n) -> (
    K := o.CoefficientRing;
    C := PP_K^(1,d);
    f := rationalMap(ring ambient C,ring PP_K^n,for i to n list random(1,ring ambient C));
    f C
);

mukaiModel = method(Options => {CoefficientRing => ZZ/65521});
mukaiModel ZZ := o -> g -> (
    K := o.CoefficientRing;
    if not member(g,{6,7,8,9,10,12}) then error "expected the genus to be in the set {6,7,8,9,10,12}";
    local psi; local X;
    if g == 6 then (
        psi = rationalMap(image(Var sub(ideal(PP_K[1,1,1]),ring PP_K^6) << PP_K^7),2,Dominant=>true);
        X = image psi;
        X.cache#"rationalParametrization" = psi;
        assert(dim X == 7 and codim X == 3 and degree X == 5 and sectionalGenus X == 1);
        return X;
    );
    if g == 7 then ( -- See [Zak - Tangents and secants of algebraic varieties - Thm. 3.8 (case 5), p. 67.]
        psi = rationalMap(image(GG(K,1,4) << PP_K^10),2,Dominant=>true);
        X = image psi;
        X.cache#"rationalParametrization" = psi;
        assert(dim X == 10 and codim X == 5 and degree X == 12 and sectionalGenus X == 7);
        return X;
    );
    if g == 8 then (-- See [Zak - Tangents and secants of algebraic varieties - Thm. 3.8 (case 3), p. 67.]
        psi = rationalMap(image(PP_K[1,1,1,1] << PP_K^8),2,Dominant=>true);
        X = image psi;
        X.cache#"rationalParametrization" = psi;
        assert(dim X == 8 and codim X == 6 and degree X == 14 and sectionalGenus X == 8);
        return X;
    );
    if g == 9 then (
        psi = rationalMap(image(PP_K^(2,2) << PP_K^6),3,2,Dominant=>true);
        X = image psi;
        X.cache#"rationalParametrization" = psi;
        assert(dim X == 6 and codim X == 7 and degree X == 16 and sectionalGenus X == 9);
        return X;
    );
    if g == 10 then ( -- p. 4 of [Kapustka and Ranestad - Vector Bundles On Fano Varieties Of Genus Ten] 
        w := gens ring PP_K^13;
        M := matrix {{0,-w_5,w_4,w_6,w_7,w_8,w_0},
                     {w_5,0,-w_3,w_12,w_13,w_9,w_1},
                     {-w_4,w_3,0,w_10,w_11,-w_6-w_13,w_2},
                     {-w_6,-w_12,-w_10,0,w_2,-w_1,w_3},
                     {-w_7,-w_13,-w_11,-w_2,0,w_0,w_4},
                     {-w_8,-w_9,w_6+w_13,w_1,-w_0,0,w_5},
                     {-w_0,-w_1,-w_2,-w_3,-w_4,-w_5,0}};
        X = projectiveVariety pfaffians(4,M);
        assert(dim X == 5 and codim X == 8 and degree X == 18 and sectionalGenus X == 10);
        psi = inverse((rationalMap {w_1,w_5,w_8,w_9,w_13,w_12})|X);
        X.cache#"rationalParametrization" = psi;        
        return X;
    );
    if g == 12 then ( -- see also pointLineAndConicOnMukaiThreefoldOfGenus12
        C := PP_K^(1,6);
        C = (rationalMap linearSpan{point ambient C,point ambient C}) C;
        psi = rationalMap(C_(random(2,C)),5,2,Dominant=>2);
        psi#"isDominant" = true;
        X = target psi;
        X.cache#"rationalParametrization" = psi;
        assert(dim X == 3 and codim X == 10 and degree X == 22 and sectionalGenus X == 12);
        return X;
    );
);

beginDocumentation()

document {Key => K3Surfaces, 
Headline => "construction of explicit examples of K3 surfaces",
"This package is for constructing explicit examples of K3 surfaces.",
References => UL{{"M. H. and G. S., ",EM"Explicit constructions of K3 surfaces and unirational Noether-Lefschetz divisors",", available at ",HREF{"https://arxiv.org/abs/2110.15819","arXiv:2110.15819"}," (2021)."}}}

document {Key => {LatticePolarizedK3surface}, 
Headline => "the class of all lattice-polarized K3 surfaces",
SeeAlso => {K3,EmbeddedK3surface}}

document {Key => {EmbeddedK3surface}, 
Headline => "the class of all embedded K3 surfaces",
SeeAlso => {LatticePolarizedK3surface,(symbol SPACE,LatticePolarizedK3surface,Sequence)}}

document {Key => {K3,(K3,ZZ,ZZ,ZZ),[K3,Verbose],[K3,CoefficientRing],[K3,Singular]}, 
Headline => "make a lattice-polarized K3 surface",
Usage => "K3(g,d,n)
K3(g,d,n,CoefficientRing=>K)", 
Inputs => {"g" => ZZ,"d" => ZZ,"n" => ZZ}, 
Outputs => {{"a ",TO2{LatticePolarizedK3surface,"K3 surface"}," defined over ",TEX///$K$///," with rank 2 lattice defined by the intersection matrix ",TEX///$\begin{pmatrix} 2g-2 & d \\ d & n \end{pmatrix}$///}}, 
EXAMPLE {"K3(6,1,-2)"},
SeeAlso => {(K3,ZZ),(K3,String),(symbol SPACE,LatticePolarizedK3surface,Sequence)}}

document {Key => {(K3,EmbeddedK3surface)}, 
Headline => "make a lattice-polarized K3 surface from an embedded K3 surface", 
Usage => "K3 S", 
Inputs => {"S" => EmbeddedK3surface =>{"a special K3 surface that contains a curve ",TEX///$C$///}}, 
Outputs => {LatticePolarizedK3surface => {"the K3 surface ",TEX///$S$///," with rank 2 lattice spanned by ",TEX///$H,C$///,", where ",TEX///$H$///," is the hyperplane section class"}}, 
EXAMPLE {"S = K3(3,5,-2);", "S(1,1)", "T = K3 S(1,1)", "T(1,0)"}}

document {Key => {(K3,String)}, 
Headline => "show available functions to construct K3 surfaces of given genus", 
Usage => "K3 \"G\"
K3(\"G\",[Unique=>true])", 
Inputs => { String => "G" => {"the string of an integer"}}, 
Outputs => {List => {"a list of terns ",TT"(g,d,n)"," such that (",TO2{(K3,ZZ,ZZ,ZZ),TT"(K3(g,d,n)"},")",TO2{(symbol SPACE,LatticePolarizedK3surface,Sequence),"(a,b)"}," is a K3 surface of genus ",TT"G",", for some integers ",TT"a,b"}}, 
EXAMPLE {"K3(\"11\",Verbose=>true)", "S = K3(5,5,-2)", "S(1,2)", "K3 S(1,2)"}, 
SeeAlso => {(K3,ZZ,ZZ,ZZ),(symbol SPACE,LatticePolarizedK3surface,Sequence)}}

undocumented {(K3,String,VisibleList)};

document {Key => {(K3,ZZ)}, 
Headline => "make a general K3 surface",
Usage => "K3 g
K3(g,CoefficientRing=>K)",
Inputs => {"g" => ZZ}, 
Outputs => {EmbeddedK3surface => {"a general K3 surface defined over ",TEX///$K$///," of genus ",TEX///$g$///," in ",TEX///$\mathbb{P}^g$///}}, 
EXAMPLE {"K3 9"},
SeeAlso => {(K3,ZZ,ZZ,ZZ)}}

document {Key => {(K3,SpecialCubicFourfold),(K3,SpecialGushelMukaiFourfold)}, 
Headline => "K3 surface associated to a cubic or GM fourfold",
Usage => "K3 X",
Inputs => {"X" => SpecialCubicFourfold => {"or ",ofClass SpecialGushelMukaiFourfold}}, 
Outputs => {EmbeddedK3surface => {"a K3 surface associated to ",TEX///$X$///}}, 
PARA {"This function calls the function ",TO associatedK3surface,"."},
EXAMPLE {"X = specialFourfold \"tau-quadric\";", "K3 X", "associatedK3surface X"},
SeeAlso => {(associatedK3surface),(K3,ZZ)}}

document {Key => {(genus,LatticePolarizedK3surface,ZZ,ZZ),(genus,EmbeddedK3surface,ZZ,ZZ)}, 
Headline => "genus of a K3 surface", 
Usage => "genus(S,a,b)", 
Inputs => {"S" => LatticePolarizedK3surface,
           "a" => ZZ,
           "b" => ZZ}, 
Outputs => {ZZ => {"the genus of ", TEX///$S$///," embedded by the complete linear system ",TEX///$|a H + b C|$///,", where ",TEX///$H,C$///," is the basis of the lattice associated to ",TEX///$S$///}}, 
EXAMPLE {"S = K3(5,2,-2)", "genus(S,2,1)"}, 
SeeAlso => {(degree,LatticePolarizedK3surface,ZZ,ZZ)}}

document {Key => {(degree,LatticePolarizedK3surface,ZZ,ZZ),(degree,EmbeddedK3surface,ZZ,ZZ)}, 
Headline => "degree of a K3 surface", 
Usage => "degree(S,a,b)", 
Inputs => {"S" => LatticePolarizedK3surface,
           "a" => ZZ,
           "b" => ZZ}, 
Outputs => {ZZ => {"the degree of ", TEX///$S$///," embedded by the complete linear system ",TEX///$|a H + b C|$///,", where ",TEX///$H,C$///," is the basis of the lattice associated to ",TEX///$S$///}}, 
EXAMPLE {"S = K3(5,2,-2)", "degree(S,2,1)"}, 
SeeAlso => {(genus,LatticePolarizedK3surface,ZZ,ZZ)}}

document {Key => {project,(project,VisibleList,LatticePolarizedK3surface,ZZ,ZZ),(project,VisibleList,EmbeddedK3surface)}, 
Headline => "project a K3 surface", 
Usage => "project({i,j,k,...},S,a,b)
project({i,j,k,...},S(a,b))", 
Inputs => {VisibleList => {"a list ",TEX///$\{i,j,k,\ldots\}$///," of nonnegative integers"},
           LatticePolarizedK3surface => "S" => {"a lattice-polarized K3 surface with rank 2 lattice spanned by ",TEX///$H,C$///},
           ZZ => "a",
           ZZ => "b"}, 
Outputs => {EmbeddedProjectiveVariety => {"the projection of ", TEX///$S$///," embedded by the complete linear system ",TEX///$|a H + b C|$///," from ",TEX///$i$///," random points of multiplicity 1, ",TEX///$j$///," random points of multiplicity 2, ",TEX///$k$///," random points of multiplicity 3, and so on until the last integer in the given list."}}, 
EXAMPLE {"S = K3(8,2,-2)", "project({5,3,1},S,2,1); -- (5th + 3rd + simple)-projection of S(2,1)"}, 
SeeAlso => {(symbol SPACE,LatticePolarizedK3surface,Sequence)}}

document {Key => {(coefficientRing,LatticePolarizedK3surface)}, 
Headline => "coefficient ring of a K3 surface", 
Usage => "coefficientRing S", 
Inputs => {"S" => LatticePolarizedK3surface}, 
Outputs => {Ring => {"the coefficient ring of ", TEX///$S$///}}, 
EXAMPLE {"K = ZZ/3331","S = K3(5,2,-2,CoefficientRing=>K)", "coefficientRing S"}}

document {Key => {(symbol SPACE,LatticePolarizedK3surface,Sequence),(symbol SPACE,EmbeddedK3surface,Sequence)}, 
Headline => "image of the embedding of a K3 surface", 
Usage => "S(a,b)", 
Inputs => {"S" => LatticePolarizedK3surface => {"a lattice-polarized K3 surface with rank 2 lattice spanned by ",TEX///$H,C$///},
          {{"a sequence of two integers ",TT"(a,b)"}}}, 
Outputs => {EmbeddedK3surface => {"the image of the embedding of ", TEX///$S$///," by the complete linear system ",TEX///$|a H + b C|$///}}, 
EXAMPLE {"S = K3(5,2,-2)", "S(1,0)", "S(2,1)"}, 
SeeAlso => {(map,LatticePolarizedK3surface,ZZ,ZZ)}}

document {Key => {(map,LatticePolarizedK3surface,ZZ,ZZ),(map,EmbeddedK3surface,ZZ,ZZ)}, 
Headline => "embedding of a K3 surface", 
Usage => "map(S,a,b)", 
Inputs => {"S" => LatticePolarizedK3surface,
           "a" => ZZ,
           "b" => ZZ}, 
Outputs => {{"the ",TO2{RationalMap,"birational morphism"}," defined by the complete linear system ",TEX///$|a H + b C|$///,", where ",TEX///$H,C$///," is the basis of the lattice associated to ",TEX///$S$///}}, 
EXAMPLE {"S = K3(3,1,-2)", "f = map(S,2,1);", "isMorphism f", "degree f", "assert(image f == S(2,1))"}, 
SeeAlso => {(symbol SPACE,LatticePolarizedK3surface,Sequence)}}

undocumented {(texMath,LatticePolarizedK3surface),(net,LatticePolarizedK3surface),(symbol ?,EmbeddedK3surface),(map,EmbeddedK3surface),(genus,EmbeddedK3surface),(degree,EmbeddedK3surface)}

document {Key => {trigonalK3,(trigonalK3,ZZ),[trigonalK3,CoefficientRing]}, 
Headline => "trigonal K3 surface", 
Usage => "trigonalK3 g", 
Inputs => {"g" => ZZ =>{"the genus"}}, 
Outputs => {LatticePolarizedK3surface => {"a random trigonal K3 surface of genus ", TEX///$g$///}}, 
PARA{"See also the paper ",EM "A remark on the generalized franchetta conjecture for K3 surfaces",", by Beauville."},
EXAMPLE {"S = trigonalK3 11", "S' = S(1,0);", "map(S',0,1)"},
SeeAlso => {(K3,ZZ,ZZ,ZZ),tetragonalK3,pentagonalK3}}

document {Key => {tetragonalK3,(tetragonalK3,ZZ),[tetragonalK3,CoefficientRing]}, 
Headline => "tetragonal K3 surface", 
Usage => "tetragonalK3 g", 
Inputs => {"g" => ZZ =>{"the genus"}}, 
Outputs => {LatticePolarizedK3surface => {"a random tetragonal K3 surface of genus ", TEX///$g$///}}, 
EXAMPLE {"S = tetragonalK3 11", "S' = S(1,0);", "map(S',0,1)"},
SeeAlso => {(K3,ZZ,ZZ,ZZ),trigonalK3,pentagonalK3}}

document {Key => {pentagonalK3,(pentagonalK3,ZZ),[pentagonalK3,CoefficientRing]}, 
Headline => "pentagonal K3 surface", 
Usage => "pentagonalK3 g", 
Inputs => {"g" => ZZ =>{"the genus"}}, 
Outputs => {LatticePolarizedK3surface => {"a random pentagonal K3 surface of genus ", TEX///$g$///}}, 
EXAMPLE {"S = pentagonalK3 11", "S' = S(1,0);", "map(S',0,1)"},
SeeAlso => {(K3,ZZ,ZZ,ZZ),trigonalK3,tetragonalK3}}

document {Key => {mukaiModel,(mukaiModel,ZZ),[mukaiModel,CoefficientRing]}, 
Headline => "Mukai models", 
Usage => "mukaiModel g", 
Inputs => {"g" => ZZ =>{"the genus"}}, 
Outputs => {EmbeddedProjectiveVariety => {"the Mukai model of genus ",TEX///$g$///," and degree ",TEX///$2g-2$///}},
EXAMPLE {"X = mukaiModel 9;", "(degree X, sectionalGenus X)", "parametrize X"}}

-- Tests --

TEST ///
for g in {3,4,5,6,7,8,9} do (
    <<"(g,d,n) = "<<(g,1,-2)<<endl;
    time S = K3(g,1,-2);
    T = S#"surface";
    C = S#"curve";
    L = S#"lattice";
    assert(dim T == 2 and degree T == 2*g-2 and sectionalGenus T == g and dim ambient T == g);
    assert(dim C == 1 and degree C == 1 and sectionalGenus C == 0 and isSubset(C,T));
    assert(L == matrix{{2*g-2,1},{1,-2}}); 
);
///

TEST ///
for e in 
select({3,4,5} ** toList(3..9),e -> ((g,d) = toSequence e; (member(g,{3,4,5}) and member(d,{3,4,5,6,7,8,9})) and (g != 5 or d <= 9) and (g != 4 or d <= 7) and (g != 3 or d <= 8)))
do (
    (g,d) := toSequence e;
    <<"(g,d,n) = "<<(g,d,0)<<endl;
    time S = K3(g,d,0);
    T = S#"surface";
    C = S#"curve";
    L = S#"lattice";
    assert(dim T == 2 and degree T == 2*g-2 and sectionalGenus T == g and dim ambient T == g);
    assert(dim C == 1 and degree C == d and sectionalGenus C == 1 and isSubset(C,T));
    assert(L == matrix{{2*g-2,d},{d,0}});    
);
///

TEST ///
for g in {3,4,5,6,7,8,9} do (
    <<"(g,d,n) = "<<(g,0,-2)<<endl;
    time S = K3(g,0,-2);
    T = S#"surface";
    C = S#"curve";
    L = S#"lattice";
    assert(dim T == 2 and degree T == 2*g-2 and sectionalGenus T == g and dim ambient T == g);
    assert(dim C == 0 and degree C == 1 and isSubset(C,T) and dim tangentSpace(T,C) > 2);
    assert(L == matrix{{2*g-2,0},{0,-2}});       
);
///

TEST ///
for g from 3 to 12 do (
    for d from 3 to 5 do (
        <<"(g,d,n) = "<<(g,d,0)<<endl;
        time S = K3(g,d,0);
        T = S#"surface";
        C = S#"curve";
        L = S#"lattice";
        assert(dim T == 2 and degree T == 2*g-2 and sectionalGenus T == g and dim ambient T == g);
        assert(dim C == 1 and degree C == d and isSubset(C,T) and sectionalGenus C == 1);
        assert(L == matrix{{2*g-2,d},{d,0}});       
    );
);
///

TEST ///
for e in 
select({3,4,5} ** toList(3..8),e -> ((g,d) = toSequence e; (member(g,{3,4,5}) and d >= 3) and (g != 5 or d <= 8) and (g != 4 or d <= 6) and (g != 3 or d <= 8)))
do (
    (g,d) := toSequence e;
    <<"(g,d,n) = "<<(g,d,-2)<<endl;
    time S = K3(g,d,-2);
    T = S#"surface";
    C = S#"curve";
    L = S#"lattice";
    assert(dim T == 2 and degree T == 2*g-2 and sectionalGenus T == g and dim ambient T == g);
    assert(dim C == 1 and degree C == d and sectionalGenus C == 0 and isSubset(C,T));
    assert(L == matrix{{2*g-2,d},{d,-2}});   
);
///

TEST /// -- randomPointedMukaiThreefold
debug K3Surfaces;
K = ZZ/333331;
for g in {3,4,5,6,7,8,9} do (
    <<"g = "<<g<<endl;
    time (X,p) = randomPointedMukaiThreefold(g,CoefficientRing=>K);
    assert(coefficientRing ring X === K and dim ambient X == g+1);
    assert isSubset(p,X);
    assert (? ideal p == "one-point scheme in PP^"|toString(g+1));
    assert(dim X == 3);
    assert(sectionalGenus X == g);
    assert(degree X == 2*g-2);
);
///

TEST /// -- randomMukaiThreefoldContainingLine
debug K3Surfaces;
K = ZZ/333331;
setRandomSeed 123456789
for g in {3,4,5,6,7,8,9} do (
    <<"g = "<<g<<endl;
    time (X,L) = randomMukaiThreefoldContainingLine(g,CoefficientRing=>K);
    assert(coefficientRing ring X === K and dim ambient X == g+1);
    assert isSubset(L,X);
    assert (? ideal L == "line in PP^"|toString(g+1));
    assert(dim X == 3);
    assert(sectionalGenus X == g);
    assert(degree X == 2*g-2);
);
///

TEST ///
for g from 3 to 7 do (
    setRandomSeed 123456789;
    <<"(g,d,n) = "<<(g,2,-2)<<endl;
    time S = K3(g,2,-2);
    T = S#"surface";
    C = S#"curve";
    L = S#"lattice";
    assert(dim T == 2 and degree T == 2*g-2 and sectionalGenus T == g and dim ambient T == g);
    assert(dim C == 1 and degree C == 2 and sectionalGenus C == 0 and isSubset(C,T));
    assert(L == matrix{{2*g-2,2},{2,-2}});
);
///

TEST ///
for g in {3,4,5,6,7,8,9} do (<<"g = "<<g<<endl; time K3 g); 
///;

end;

-- Hard tests

TEST ///
K3 10;
K3 12;
///

TEST ///
K3 11 
///

TEST ///
for g from 8 to 12 do (
    setRandomSeed 123456789;
    <<"(g,d,n) = "<<(g,2,-2)<<endl;
    time S = K3(g,2,-2);
    T = S#"surface";
    C = S#"curve";
    L = S#"lattice";
    assert(dim T == 2 and degree T == 2*g-2 and sectionalGenus T == g and dim ambient T == g);
    assert(dim C == 1 and degree C == 2 and sectionalGenus C == 0 and isSubset(C,T));
    assert(L == matrix{{2*g-2,2},{2,-2}});
);
///

TEST /// -- randomMukaiThreefoldContainingLine
debug K3Surfaces;
K = ZZ/333331;
for g in {10,12} do (
    setRandomSeed 12345;
    <<"g = "<<g<<endl;
    time (X,L) = randomMukaiThreefoldContainingLine(g,CoefficientRing=>K);
    assert(coefficientRing ring X === K and dim ambient X == g+1);
    assert isSubset(L,X);
    assert (? ideal L == "line in PP^"|toString(g+1));
    assert(dim X == 3);
    assert(sectionalGenus X == g);
    assert(degree X == 2*g-2);
);
///

TEST /// -- randomPointedMukaiThreefold
debug K3Surfaces;
K = ZZ/333331;
for g in {10,12} do (
    <<"g = "<<g<<endl;
    time (X,p) = randomPointedMukaiThreefold(g,CoefficientRing=>K);
    assert(coefficientRing ring X === K and dim ambient X == g+1);
    assert isSubset(p,X);
    assert (? ideal p == "one-point scheme in PP^"|toString(g+1));
    assert(dim X == 3);
    assert(sectionalGenus X == g);
    assert(degree X == 2*g-2);
);
///

TEST ///
for g from 13 to 15 do (
    for d from 3 to 5 do (
        <<"(g,d,n) = "<<(g,d,0)<<endl;
        time S = K3(g,d,0);
        T = S#"surface";
        C = S#"curve";
        L = S#"lattice";
        assert(dim T == 2 and degree T == 2*g-2 and sectionalGenus T == g and dim ambient T == g);
        assert(dim C == 1 and degree C == d and isSubset(C,T) and sectionalGenus C == 1);
        assert(L == matrix{{2*g-2,d},{d,0}});       
    );
);
///

TEST ///
setRandomSeed 123456789;
for g in {10,12} do (
    <<"(g,d,n) = "<<(g,1,-2)<<endl;
    time S = K3(g,1,-2);
    T = S#"surface";
    C = S#"curve";
    L = S#"lattice";
    assert(dim T == 2 and degree T == 2*g-2 and sectionalGenus T == g and dim ambient T == g);
    assert(dim C == 1 and degree C == 1 and sectionalGenus C == 0 and isSubset(C,T));
    assert(L == matrix{{2*g-2,1},{1,-2}}); 
);
///

TEST ///
for g in {10,12} do (
    <<"(g,d,n) = "<<(g,0,-2)<<endl;
    time S = K3(g,0,-2);
    T = S#"surface";
    C = S#"curve";
    L = S#"lattice";
    assert(dim T == 2 and degree T == 2*g-2 and sectionalGenus T == g and dim ambient T == g);
    assert(dim C == 0 and degree C == 1 and isSubset(C,T) and dim tangentSpace(T,C) > 2);
    assert(L == matrix{{2*g-2,0},{0,-2}});       
);
///

TEST ///
K3 20
///

TEST ///
K3 22
///