arXiv subjects

Anatoli Shatsila

Publications and source records attributed to Anatoli Shatsila.

10 recordsLinked to original sources

arXiv2026

Theta-duality and Prym-Torelli for cyclic covers of hyperelliptic curves

Let $f:\tilde{C}\to C$ be an étale cyclic cover of odd prime degree $d$ of a hyperelliptic curve of genus $g\geq 2$. We revisit the theta-duality reconstruction argument for the generic injectivity of the associated Prym map by Naranjo-Ortega-Pirola-Spelta and identify an additional residual locus arising from the fixed divisor of the line bundles occurring in that argument. For $d\ge5$, this locus does not affect reconstruction, giving injectivity under the numerical assumption $(d-1)(g-1)\geq 7$. For $d=3$, the geometry is governed instead by trigonal pencils on the quotient of $\tilde{C}$ by a lift of the hyperelliptic involution; this yields generic degree $2$ in genus $5$ and injectivity for $g\ge6$.

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arXiv2026

Finiteness and injectivity of Prym maps for cyclic coverings

The structure of the Prym map for coverings of degrees $d\geq 3$ is mostly unknown. Only recently, under mild numerical assumptions, a generic injectivity of the Prym maps for étale cyclic coverings of hyperelliptic curves of prime degrees has been shown. In the paper, we prove that the Prym maps is generically injective for all remaining degrees (i.e. composite numbers $d\geq 6$) and we prove global injectivity if $d$ is not a power of an odd prime. In particular, we complete the study of Prym maps of étale cyclic coverings of genus 2 curves. As an application, we fully characterise for which $d$ the Prym map of cyclic coverings of degree $d$ of genus $3$ curves is generically finite and we conjecture that it is injective.

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arXiv2026

A note on smooth quotients of Prym varieties

We study pseudoreflections of geometric origin on Prym varieties of étale double covers. We prove that if the genus of the base curve is $g \geq 4$ then every such pseudoreflection has order 2. We use this result to show that, for $g \geq 5$, a non-trivial finite group $G$ of automorphisms of geometric origin acting faithfully on the Prym $P$ with $P/G$ smooth must be isomorphic to either $\mathbb{Z}/2\mathbb{Z}$ or $(\mathbb{Z}/2\mathbb{Z})^2$. We also show that the latter case can occur only for $g \leq 7$. This sharpens results of Auffarth, Lahoz and Naranjo.

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arXiv2026

On the Prym map of degree 4 cyclic covers of hyperelliptic curves

In this paper, we study the Prym map associated to degree 4 étale cyclic covers of genus $g$ hyperelliptic curves restricted to the irreducible component $\mathcal{RH}_g[4]^{hyp}$ of the moduli space of such covers where an intermediate cover is hyperelliptic. We show that for $g \geq 3$ the Prym map is injective on $\mathcal{RH}_g[4]^{hyp}$. In the case $g=2$ (where $\mathcal{RH}_2[4]^{hyp} = \mathcal{RH}_2[4]$) we prove that non-empty fibers of the Prym map, apart from two exceptional fibers, are isomorphic to the projective line without 8 points. Moreover, we obtain a new description of the space $\mathcal{RH}_g[4]^{hyp}$ in terms of tuples of complex numbers and find equations of hyperelliptic curves arising from such covers.

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arXiv2026

Pryms of $\mathbb{Z}_3\times\mathbb{Z}_3$ coverings of genus 2 curves

We study unramified Galois $\mathbb{Z}_3 \times \mathbb{Z}_3$ coverings of genus 2 curves and the corresponding Prym varieties and Prym maps. In particular, we prove that any such covering can be reconstructed from its Prym variety, that is, the Prym-Torelli theorem holds for these coverings. We also investigate the Prym map of unramified $G$-coverings of genus 2 curves for an arbitrary abelian group $G$. We show that the generic fiber of the Prym map is finite unless $G$ is cyclic of order less than 6

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arXiv2025

Characteristic numbers of algebras

We introduce characteristic numbers of a finite commutative unital $\mathbb{C}$-algebra, which are numerical invariants arising from algebraic intersection theory. We characterize Gorenstein and local complete intersection algebras in terms of their characteristic numbers. We compute characteristic numbers for certain families of algebras. We show that characteristic numbers are constant on $\mathrm{Hilb}_d(\mathbb{A}^1)$, provide an explicit upper bound for characteristic numbers on the smoothable component of $\mathrm{Hilb}_d(\mathbb{A}^n)$ and an explicit lower bound for characteristic numbers on the Gorenstein locus of $\mathrm{Hilb}_d(\mathbb{A}^n)$ for $n \geq d-2$.

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arXiv2025

On the construction of a counterexample to Strassen's rank additivity conjecture

The rank additivity conjecture, first formulated by Volker Strassen in 1973, states that the rank of the direct sum of two independent tensors is equal to the sum of their individual ranks. In the last decades, this conjecture has been a central topic in tensor rank theory and its implications for computational complexity. In 2019, Yaroslav Shitov disproved this conjecture in its general form by showing the existence of a counter-example using a dimension counting argument. In this paper, we provide an overview of the Strassen problem and Shitov's work and revisit his counterexample with a detailed explanation, offering an alternative proof.

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arXiv2024

Exact values of generic subrank

In this article we prove the subrank of a generic tensor in $\mathbb{C}^{n,n,n}$ to be $Q(n) = \lfloor\sqrt{3n - 2}\rfloor$ by providing a lower bound to the known upper bound. More generally, we find the generic subrank of tensors of all orders and dimensions. This answers two open questions posed in arXiv:2205.15168v2. Finally, we compute dimensions of varieties of tensors of subrank at least $r$.

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arXiv2023

Hyperelliptic genus 3 curves with involutions and a Prym map

We characterise genus 3 complex smooth hyperelliptic curves that contain two additional involutions as curves that can be build from five points in $\mathbb{P}^1$ with a distinguished triple. We are able to write down explicit equations for the curves and all their quotient curves. We show that, fixing one of the elliptic quotient curve, the Prym map becomes a 2:1 map and therefore the hyperelliptic Klein Prym map, constructed recently by the first author with A. Ortega, is also 2:1 in this case. As a by-product we show an explicit family of $(1, d)$ polarised abelian surfaces (for d > 1), such that any surface in the family satisfying a certain explicit condition is abstractly non-isomorphic to its dual abelian surface.

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arXiv2022

Geometry of elliptic normal curves of degree 6

In our work we focus on the geometry of elliptic normal curves of degree 6 embedded in $\mathbb{P}^5$. We determine the space of quadric hypersurfaces through an elliptic normal curve of degree 6 and find the explicit equations of generators of $I(\text{Sec}(C_6))$. We study the images $C_p$ and $C_{pq}$ of a sextic $C_6$ under the projection from a general point $P \in \mathbb{P}^5$ and a general line $\overline{PQ} \subset \mathbb{P}^5$. In particular, we show that $C_p$ is $k$-normal for all $k \geq 2$ and $I(C_p)$ is generated by three homogeneous polynomials of degree 2 and two homogeneous polynomials of degree 3. We then show that $C_{pq}$ is $k$-normal for all $k \geq 3$ and $I(C_{pq})$ is generated by two homogeneous polynomials of degree 3 and three homogeneous polynomials of degree 4.

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