Explicit constructions of K3 surfaces and unirational Noether–Lefschetz divisors

Hoff, Michael; Staglianò, Giovanni

doi:10.1016/j.jalgebra.2022.08.005

Cited by 5 publications

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References 28 publications

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“…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%

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%

The SpecialFanoFourfolds package in Macaulay2

Staglianò

2024

J. Softw. Alg. Geom.

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Curves on Brill–Noether special K3 surfaces

Haburcak

2024

Mathematische Nachrichten

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Mukai showed that projective models of Brill–Noether general polarized K3 surfaces of genus 6–10 and 12 are obtained as linear sections of projective homogeneous varieties, and that their hyperplane sections are Brill–Noether general curves. In general, the question, raised by Knutsen, and attributed to Mukai, of whether the Brill–Noether generality of any polarized K3 surface is equivalent to the Brill–Noether generality of smooth curves in the linear system , is still open. Using Lazarsfeld–Mukai bundle techniques, we answer this question in the affirmative for polarized K3 surfaces of genus , which provides a new and unified proof even in the genera where Mukai models exist.

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Lifts of line bundles on curves on K3 surfaces

Watanabe,

Komeda

2024

Abh. Math. Semin. Univ. Hambg.

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Explicit constructions of K3 surfaces and unirational Noether–Lefschetz divisors

Cited by 5 publications

References 28 publications

The SpecialFanoFourfolds package in Macaulay2

The SpecialFanoFourfolds package in Macaulay2

Curves on Brill–Noether special K3 surfaces

Lifts of line bundles on curves on K3 surfaces

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