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Packages ┬╗ SpecialFanoFourfolds :: specialGushelMukaiFourfold(Array,Array,String,Thing)
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specialGushelMukaiFourfold(Array,Array,String,Thing) -- construct GM fourfolds by gluing cubic or quartic scrolls to surfaces in PP^6



From the returned fourfold $X$, with the following commands we obtain the surface $S$, the curve $C$, and the scroll $B$ used in the construction:

(B,C) = X.cache#"Construction"; S = ambientVariety C;

Then the surface $\overline{\psi_{B}(S)}\subset\mathbb{G}(1,4)$ can be constructed with

psi = rationalMap B; (psi S)%(image psi);

In the following example we construct a GM fourfold containing the image via $\psi_B:\mathbb{P}^6\dashrightarrow\mathbb{G}(1,4)$ of a quintic del Pezzo surface $S\subset\mathbb{P}^5\subset\mathbb{P}^6$, obtained as the image of the plane via the linear system of quartic curves with three general simple base points and two general double points, which cuts $B\simeq\mathbb{P}^1\times\mathbb{P}^2\subset\mathbb{P}^5\subset\mathbb{P}^6$ along a rational normal quartic curve obtained as the image of a general conic passing through the two double points.

i1 : X = specialGushelMukaiFourfold([4, 3, 2],[2, 0, 2]);

o1 : ProjectiveVariety, GM fourfold containing a surface of degree 6 and sectional genus 1
i2 : describe X

o2 = Special Gushel-Mukai fourfold of discriminant 18(')
     containing a surface in PP^8 of degree 6 and sectional genus 1
     cut out by 11 hypersurfaces of degrees (1,1,2,2,2,2,2,2,2,2,2)
     and with class in G(1,4) given by 3*s_(3,1)+3*s_(2,2)
     Type: ordinary
i3 : (B,C) = X.cache#"Construction";
i4 : S = ambientVariety C;

o4 : ProjectiveVariety, surface in PP^6
i5 : C;

o5 : ProjectiveVariety, curve in PP^6 (subvariety of codimension 1 in S)
i6 : B;

o6 : ProjectiveVariety, threefold in PP^6
i7 : assert(C == S * B)


See also

Ways to use this method: