The SpecialFanoFourfolds package in Macaulay2

Abstract

We introduce the Macaulay2 package SpecialFanoFourfolds, a package that provides functions for working with cubic fourfolds, Gushel-Mukai fourfolds, and some other special Fano fourfolds. INTRODUCTION. The package SpecialFanoFourfolds in Macaulay2[2] provides support for some types of Hodge-special Fano fourfolds, which can be represented as smooth hypersurfaces of low degree r in some fixed ambient fivefold ‫.ޖ‬ Roughly speaking, a smooth degree-r hypersurface X ⊂ ‫ޖ‬ is called Hodge-special if it contains an algebraic surface S whose cohomology class does not come from the ambient fivefold ‫.ޖ‬ In the parameter space of all smooth hypersurfaces X ⊂ ‫ޖ‬ of degree r , the locus of Hodge-special fourfolds is called the Noether-Lefschetz locus.One important case is that of smooth cubic hypersurfaces in ‫ސ‬ 5 (cubic fourfolds for short). The Noether-Lefschetz locus in the 20-dimensional moduli space C = (‫(ސ‬O ‫ސ‬ 5 (3)) \ Disc 3 ‫ސ‬ 5 )/PGL 6 of cubic fourfolds is a countable union of irreducible hypersurfaces C d ⊂ C, where d > 6 with d ≡ 0, 2 mod 6. The hypersurface C d parametrizes cubic fourfolds of discriminant d, that is, the set of cubic fourfolds X which contain a surface S such that the discriminant of the saturated lattice spanned by h 2 X and [S] in H 2,2 (X, ‫)ޚ‬ := H 4 (X, ‫)ޚ‬ ∩ H 2 ( 2 X ) is d (here h X stands for the class of a hyperplane section of X ). For general results on cubic fourfolds, we refer the reader to [3; 4].Another important case is that of smooth quadric hypersurfaces in a 5-dimensional linear section ‫ޖ‬ ⊂ ‫ސ‬ 8 of the cone in ‫ސ‬ 10 over the Grassmannian ‫,1(އ‬ 4) ⊂ ‫ސ‬ 9 . Such hypersurfaces in ‫ޖ‬ are known as Gushel-Mukai fourfolds (GM fourfolds for short), and are parametrized by a moduli space GM of dimension 24. Outside a closed subset of codimension 2 in GM, we have that the ambient fivefold ‫ޖ‬ is smooth, and hence it is isomorphic to a hyperplane section of ‫,1(އ‬ 4) ⊂ ‫ސ‬ 9 . In such case, a GM fourfold X ⊂ ‫ޖ‬ is called ordinary. The Noether-Lefschetz locus in GM is a countable union of hypersurfaces GM d , labeled by the possible values of the discriminant d, which are the integers d > 8 with d ≡ 0, 2, 4 mod 8. If d ≡ 2 mod 8, then GM d is the union of two irreducible components GM ′ d ∪ GM ′′ d ; otherwise it is irreducible. For general results on Gushel-Mukai fourfolds, we refer the reader to [1].