Orlov's equivalence and maximal Cohen‐Macaulay modules over the cone of an elliptic curve

Galinat, Lennart

doi:10.1002/mana.201300106

Cited by 7 publications

(8 citation statements)

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“…Classification of rank 1 MCM modules over the general Hesse cubic x 3 0 + x 3 1 + x 3 2 − 3ψx 0 x 1 x 2 = 0 follows immediately from our results, where the only non-trivial case is given by Moore's matrices. Analogous classification for rank 1 MCM modules over Weierstrass cubics y 2 z − x 3 − axz − bz 3 = 0 here obtained more recently in [18].…”

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Betti tables of MCM modules over the cone of a plane cubic

Pavlov

2020

Math. Z.

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We show that for maximal Cohen-Macaulay modules over a homogeneous coordinate rings of smooth Calabi-Yau varieties X computation of Betti numbers can be reduced to computations of dimensions of certain Hom groups in the bounded derived category D b (X).In the simplest case of a smooth elliptic curve E imbedded into P 2 as a smooth cubic we use our formula to get explicit answers for Betti numbers. Description of the automorphism group of the derived category D b (E) in terms of the spherical twist functors of Seidel and Thomas plays a major role in our approach. We show that there are only four possible shapes of the Betti tables up to a shifts in internal degree, and two possible shapes up to a shift in internal degree and taking syzygies.Determinantal presentations of smooth cubics were also studied in [12]. Where authors obtained an explicit solution for the Legendre form of the cubic.Key ingredient of our treatment of Betti numbers of graded modules over R E is an equivalence of triangulated categoriesproved by D.Orlov in [24]. In this paper we call this equivalence of categories Orlov's equivalence. We explain notations and provide some details in the next section. This equivalence is used to formulate questions about graded MCM modules over R E in terms of coherent sheaves on E .The first result (see theorem 3.2 below) in this direction is immediate.

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confidence: 77%

“…A complete description of matrix factorizations of rank one MCM modules over the Fermat cubic can be found in [23], and that for rank one MCM modules over a general elliptic curve in Weierstrass form in [18].…”

Section: Examples Of Matrix Factorizations Of a Smooth Cubicmentioning

confidence: 99%

Betti tables of MCM modules over the cone of a plane cubic

Pavlov

2020

Math. Z.

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“…Studying linear determinantal representations is a classical topic in algebraic geometry [1,2]. Recently it also appears in the study of derived categories [3], semidefinite programming [4]. There are also studies from arithmetic viewpoints [5][6][7][8].…”

Section: Introductionmentioning

confidence: 99%

On algorithms to obtain linear determinantal representations of smooth plane curves of higher degree

2019

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We give two algorithms to compute linear determinantal representations of smooth plane curves of any degree over any field. As particular examples, we explicitly give representatives of all equivalence classes of linear determinantal representations of two special quartics over the field Q of rational numbers, the Klein quartic and the Fermat quartic. Keywords linear determinantal representations, plane quartics, the Klein quartic, the Fermat quarticResearch Activity Group Algorithmic Number Theory and Its Applications

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“…Studying linear determinantal representations of smooth plane cubics is a classical topic in linear algebra and algebraic geometry ( [Vin89], [Dol12]). Recently, they appear in the study of the derived category of smooth plane cubics ( [Gal14]) and the theory of space-time codes ( [DG08]). They have been studied from arithmetic viewpoints ( [FN14], [II16a], [Ish15]).…”

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confidence: 99%

An algorithm to obtain linear determinantal representations of smooth plane cubics over finite fields

Ishitsuka¹

2016

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We give a brief report on our computations of linear determinantal representations of smooth plane cubics over finite fields. After recalling a classical interpretation of linear determinantal representations as rational points on the affine part of Jacobian varieties, we give an algorithm to obtain all linear determinantal representations up to equivalence. We also report our recent study on computations of linear determinantal representations of twisted Fermat cubics defined over the field of rational numbers. This paper is a summary of the author's talk at the JSIAM JANT workshop on algorithmic number theory in March, 2016. Details will appear elsewhere.

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Orlov's equivalence and maximal Cohen‐Macaulay modules over the cone of an elliptic curve

Cited by 7 publications

References 26 publications

Betti tables of MCM modules over the cone of a plane cubic

Betti tables of MCM modules over the cone of a plane cubic

On algorithms to obtain linear determinantal representations of smooth plane curves of higher degree

An algorithm to obtain linear determinantal representations of smooth plane cubics over finite fields

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