Explicit Computations with Cubic Fourfolds, Gushel–Mukai Fourfolds, and their Associated K3 Surfaces

Staglianò, Giovanni

doi:10.1080/10586458.2023.2184882

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“…Particularly important is the case when the value of (1-1) is 1 since in this case we generally obtain a geometric description of a whole irreducible component of the Noether-Lefschetz locus. See [11] and [12] for a systematic use of the function parameterCount to deduces some birational properties of the first components of the Noether-Lefschetz locus in the moduli space of GM fourfolds; see also [7].…”

Section: The Main Functions Of the Packagementioning

confidence: 99%

“…Then the inverse map of µ| X is defined by a linear system of hypersurfaces in W with points of multiplicity e along a surface U ⊂ W , which turns out to be a projection of a K3 surface U ⊂ ‫ސ‬ g of degree d and genus g = d/2 + 1. For details and precise results, see [10]; see also [6] and [12]. The function associatedK3surface applied to the fourfold X returns this surface U .…”

Section: The Main Functions Of the Packagementioning

confidence: 99%

“…As an example, we now compute the K3 surface U ⊂ ‫ސ‬ 6 associated with the GM fourfolds X containing a σ 3,1 -plane. The function associatedK3surface can be used to construct explicit general K3 surfaces of given genus g. Currently, this can be done for g ∈ {6, 8, 11, 14, 20, 22}, and consequently for these values we get an explicit unirationality of the moduli space F g of polarized K3 surfaces of genus g. See [12] and also [6; 10] for more explanations and references on this topic.…”

Section: The Main Functions Of the Packagementioning

confidence: 99%

“…In the following section, we give just a brief introduction to how to use the package. For more computational details and recent applications of the package, see [12].…”

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The SpecialFanoFourfolds package in Macaulay2

Staglianò

2024

J. Softw. Alg. Geom.

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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].

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Section: The Main Functions Of the Packagementioning

confidence: 99%

Section: The Main Functions Of the Packagementioning

confidence: 99%

Section: The Main Functions Of the Packagementioning

confidence: 99%

“…In the following section, we give just a brief introduction to how to use the package. For more computational details and recent applications of the package, see [12].…”

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