# surface(MultiprojectiveVariety,MultiprojectiveVariety) -- make a Hodge-special surface

## Synopsis

• Function: surface
• Usage:
surface(C,S)
• Inputs:
• C, , an irreducible curve
• S, , a smooth surface $S$ containing the curve $C$
• Outputs:
• the Hodge special surface corresponding to the pair $(C,S)$

## Description

The curve $C$ can be recovered using the function curve.

 i1 : K = ZZ/65521; i2 : C = random PP_K^(1,3); -- random twisted cubic in P^3 o2 : ProjectiveVariety, curve in PP^3 i3 : j = parametrize PP_K(1,1,1,4); o3 : WeightedRationalMap (birational map from PP^3 to PP(1,1,1,4)) i4 : C = (rationalMap(ambient C,source j) * j) C; o4 : ProjectiveVariety, curve in PP(1,1,1,4) i5 : describe C o5 = ambient:.............. PP(1,1,1,4) dim:.................. 1 codim:................ 2 degree:............... 12 generators:........... 3^1 5^2 8^1 purity:............... true dim sing. l.:......... -1 i6 : S = random(8,C); o6 : ProjectiveVariety, surface in PP(1,1,1,4) i7 : describe S o7 = ambient:.............. PP(1,1,1,4) dim:.................. 2 codim:................ 1 degree:............... 8 generators:........... 8^1 purity:............... true dim sing. l.:......... -1 i8 : S = surface(C,S); o8 : ProjectiveVariety, octic surface in PP(1,1,1,4) with rank 2 lattice defined by the intersection matrix | 2 3 | (det: -19) | 3 -5 | i9 : discriminant S o9 = -19 i10 : parameterCount(S,Verbose=>true) C: curve in PP(1,1,1,4) cut out by 4 hypersurfaces of degrees 3^1 5^2 8^1 S: surface in PP(1,1,1,4) defined by a form of degree 8 ambient: P = PP(1,1,1,4) h^1(N_{C,P}) = 1 --warning: condition h^1(N_{C,P}) == 0 not satisfied h^0(N_{C,P}) = 21 h^0(I_{C,P}(8)) = 36 h^0(N_{C,P}) + 35 = 56 h^0(N_{C,S}) = 0 dim{[S] : C ⊂ S ⊂ P} >= 56 dim P(H^0(O_P(8))) = 60 codim{[S] : C ⊂ S ⊂ P} <= 4 o10 = (4, (36, 21, 0)) o10 : Sequence i11 : f := map(S,1,0) o11 = multi-rational map consisting of one single rational map source variety: surface in PP(1,1,1,4) defined by a form of degree 8 target variety: PP^2 o11 : WeightedRationalMap (rational map from S to PP^2) i12 : f = quadricFibration f o12 = multi-rational map consisting of one single rational map source variety: surface in PP(1,1,1,4) defined by a form of degree 8 target variety: PP^2 o12 : QuadricFibration (rational map from S to PP^2) i13 : discriminant f -- starting computation of the generic fiber... -- computation of the generic fiber successfully completed. -- verifying the computation of the discriminant locus o13 = curve in PP^2 defined by a form of degree 8 o13 : ProjectiveVariety, curve in PP^2

## Caveat

This feature is currently under development.