G-invariant Hilbert schemes on Abelian surfaces and enumerative geometry of the orbifold Kummer surface

AbstractThe aim of this paper is to describe the geometry of the generic Kummer surface associated to a (1, 2)-polarized abelian surface. We show that it is the double cover of a weak del Pezzo surface and that it inherits from the del Pezzo surface an interesting elliptic fibration with twelve singular fibers of typeI2.

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Kummer surfaces associated to (1, 2)-polarized abelian surfaces

Nagoya Mathematical Journal ◽

10.1215/00277630-1260477 ◽

2011 ◽

Vol 202 ◽

pp. 127-143 ◽

Cited By ~ 1

Author(s):

Afsaneh Mehran

Keyword(s):

Double Cover ◽

Pezzo Surface ◽

Abelian Surface ◽

Kummer Surface ◽

Abelian Surfaces ◽

Del Pezzo Surface ◽

Singular Fibers ◽

Kummer Surfaces ◽

Del Pezzo

AbstractThe aim of this paper is to describe the geometry of the generic Kummer surface associated to a (1, 2)-polarized abelian surface. We show that it is the double cover of a weak del Pezzo surface and that it inherits from the del Pezzo surface an interesting elliptic fibration with twelve singular fibers of type I2.

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Constructing abelian surfaces for cryptography via Rosenhain invariants

LMS Journal of Computation and Mathematics ◽

10.1112/s1461157014000370 ◽

2014 ◽

Vol 17 (A) ◽

pp. 157-180 ◽

Cited By ~ 4

Author(s):

Craig Costello ◽

Alyson Deines-Schartz ◽

Kristin Lauter ◽

Tonghai Yang

Keyword(s):

Complex Multiplication ◽

Rational Functions ◽

The Other ◽

Algebraic Numbers ◽

Kummer Surface ◽

Abelian Surfaces ◽

Minimal Polynomials ◽

Genus 2 ◽

Kummer Surfaces ◽

Security Levels

AbstractThis paper presents an algorithm to construct cryptographically strong genus $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2$ curves and their Kummer surfaces via Rosenhain invariants and related Kummer parameters. The most common version of the complex multiplication (CM) algorithm for constructing cryptographic curves in genus 2 relies on the well-studied Igusa invariants and Mestre’s algorithm for reconstructing the curve. On the other hand, the Rosenhain invariants typically have much smaller height, so computing them requires less precision, and in addition, the Rosenhain model for the curve can be written down directly given the Rosenhain invariants. Similarly, the parameters for a Kummer surface can be expressed directly in terms of rational functions of theta constants. CM-values of these functions are algebraic numbers, and when computed to high enough precision, LLL can recognize their minimal polynomials. Motivated by fast cryptography on Kummer surfaces, we investigate a variant of the CM method for computing cryptographically strong Rosenhain models of curves (as well as their associated Kummer surfaces) and use it to generate several example curves at different security levels that are suitable for use in cryptography.

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A Fourier-Mukai approach to the enumerative geometry of principally polarized abelian surfaces

Mathematische Nachrichten ◽

10.1002/mana.201100182 ◽

2012 ◽

Vol 285 (16) ◽

pp. 1981-1998

Author(s):

Antony Maciocia

Keyword(s):

Enumerative Geometry ◽

Abelian Surfaces

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Universal functors on symmetric quotient stacks of Abelian varieties

Selecta Mathematica ◽

10.1007/s00029-021-00740-4 ◽

2021 ◽

Vol 28 (2) ◽

Author(s):

Andreas Krug ◽

Ciaran Meachan

Keyword(s):

Elliptic Curve ◽

Abelian Varieties ◽

Abelian Surface ◽

Hilbert Schemes ◽

Abelian Surfaces ◽

The One ◽

Dimension One

AbstractWe consider certain universal functors on symmetric quotient stacks of Abelian varieties. In dimension two, we discover a family of $${{\mathbb {P}}}$$ P -functors which induce new derived autoequivalences of Hilbert schemes of points on Abelian surfaces; a set of braid relations on a holomorphic symplectic sixfold; and a pair of spherical functors on the Hilbert square of an Abelian surface, whose twists are related to the well-known Horja twist. In dimension one, our universal functors are fully faithful, giving rise to a semiorthogonal decomposition for the symmetric quotient stack of an elliptic curve (which we compare to the one discovered by Polishchuk–Van den Bergh), and they lift to spherical functors on the canonical cover, inducing twists which descend to give new derived autoequivalences here as well.

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Principal polarizations of supersingular abelian surfaces

Journal of the Mathematical Society of Japan ◽

10.2969/jmsj/82528252 ◽

2020 ◽

Vol 72 (4) ◽

pp. 1161-1180

Author(s):

Tomoyoshi IBUKIYAMA

Keyword(s):

Abelian Surfaces

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Curved Rickard complexes and link homologies

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2019-0044 ◽

2020 ◽

Vol 2020 (769) ◽

pp. 87-119

Author(s):

Sabin Cautis ◽

Aaron D. Lauda ◽

Joshua Sussan

Keyword(s):

Spectral Sequence ◽

Quantum Groups ◽

Braid Group ◽

Derived Categories ◽

Duke Math ◽

Khovanov Homology ◽

Hilbert Schemes ◽

Coherent Sheaves ◽

Knot Homology ◽

Original Motivation

AbstractRickard complexes in the context of categorified quantum groups can be used to construct braid group actions. We define and study certain natural deformations of these complexes which we call curved Rickard complexes. One application is to obtain deformations of link homologies which generalize those of Batson–Seed [3] [J. Batson and C. Seed, A link-splitting spectral sequence in Khovanov homology, Duke Math. J. 164 2015, 5, 801–841] and Gorsky–Hogancamp [E. Gorsky and M. Hogancamp, Hilbert schemes and y-ification of Khovanov–Rozansky homology, preprint 2017] to arbitrary representations/partitions. Another is to relate the deformed homology defined algebro-geometrically in [S. Cautis and J. Kamnitzer, Knot homology via derived categories of coherent sheaves IV, colored links, Quantum Topol. 8 2017, 2, 381–411] to categorified quantum groups (this was the original motivation for this paper).

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The close relation between border and Pommaret marked bases

Collectanea mathematica ◽

10.1007/s13348-021-00313-w ◽

2021 ◽

Author(s):

Cristina Bertone ◽

Francesca Cioffi

Keyword(s):

Finite Order ◽

Polynomial Ring ◽

Group Actions ◽

Close Relation ◽

Open Covering ◽

Order Ideal ◽

Lower Dimension ◽

Hilbert Schemes ◽

Affine Spaces ◽

The Ideal

AbstractGiven a finite order ideal $${\mathcal {O}}$$ O in the polynomial ring $$K[x_1,\ldots , x_n]$$ K [ x 1 , … , x n ] over a field K, let $$\partial {\mathcal {O}}$$ ∂ O be the border of $${\mathcal {O}}$$ O and $${\mathcal {P}}_{\mathcal {O}}$$ P O the Pommaret basis of the ideal generated by the terms outside $${\mathcal {O}}$$ O . In the framework of reduction structures introduced by Ceria, Mora, Roggero in 2019, we investigate relations among $$\partial {\mathcal {O}}$$ ∂ O -marked sets (resp. bases) and $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked sets (resp. bases). We prove that a $$\partial {\mathcal {O}}$$ ∂ O -marked set B is a marked basis if and only if the $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked set P contained in B is a marked basis and generates the same ideal as B. Using a functorial description of these marked bases, as a byproduct we obtain that the affine schemes respectively parameterizing $$\partial {\mathcal {O}}$$ ∂ O -marked bases and $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked bases are isomorphic. We are able to describe this isomorphism as a projection that can be explicitly constructed without the use of Gröbner elimination techniques. In particular, we obtain a straightforward embedding of border schemes in affine spaces of lower dimension. Furthermore, we observe that Pommaret marked schemes give an open covering of Hilbert schemes parameterizing 0-dimensional schemes without any group actions. Several examples are given throughout the paper.

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