arXiv subjects

Paweł Borówka

Publications and source records attributed to Paweł Borówka.

10 recordsLinked to original sources

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.

math.AG↗

arXiv2026

Klein coverings over hyperelliptic genus 3 curves

We characterize the moduli space of étale Klein coverings (i.e. Galois with deck group $\mathbb{Z}_2^2$) of hyperelliptic curves of genus 3. We prove that the Prym map on each component is injective. As an application, we show that the Prym map of étale Klein coverings of genus 3 curves is generically finite.

math.AG↗

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

math.AG↗

arXiv2024

Involutions on hyperelliptic curves and Prym maps

We investigate the geometry of smooth hyperelliptic curves that possess additional involutions, especially from the point of view of the Prym theory. Our main result is the injectivity of the Prym map for hyperelliptic $\mathbb{Z}_2^2$-coverings over hyperelliptic curves of positive genus.

math.AG↗

arXiv2024

A note on dual abelian varieties

For any non-principal polarisation $D$, we explicitly construct $D$-polarised abelian variety $A$, such that its dual abelian variety is not (abstractly) isomorphic to $A$. For $\dim(A)>3$ the construction includes examples with submaximal Picard number equal to $(\dim(A)-1)^2+1$. As a corollary, we show that a very general non-principally polarised abelian variety is not isomorphic to its dual. Moreover, we show an example of an abelian variety that is isomorphic to its dual, yet it does not admit a principal polarisation.

math.AG↗

arXiv2022

Non-simple polarised abelian surfaces and genus 3 curves with completely decomposable Jacobians

We study the space of non-simple polarised abelian surfaces. Specifically, we describe for which pairs $(m,n)$ the locus of polarised abelian surfaces of type $(1,d)$ that contain two complementary elliptic curve of exponents $m,n$, denoted $\mathcal{E}_d(m,n)$ is non-empty. We show that if $d$ is square-free, the locus $\mathcal{E}_d(m,n)$ is an irreducible surface (if non-empty). We also show that the loci $\mathcal{E}_d(d,d)$ can have many components if $d$ is an odd square. As an application, we show that for a genus $3$ curve with a completely decomposable Jacobian (i.e. isogenous to a product of 3 elliptic curves) the degrees of complementary coverings $f_i:C\rightarrow E_i,\ i=1,2,3$ satisfy $lcm(deg(f_1),deg(f_2))=lcm(deg(f_1),deg(f_3))=lcm(deg(f_2),deg(f_3))$.

math.AG↗

arXiv2019

Klein coverings of genus 2 curves

We investigate the geometry of étale $4:1$ coverings of smooth complex genus 2 curves with the monodromy group isomorphic to the Klein four-group. There are two cases, isotropic and non-isotropic depending on the values of the Weil pairing restricted to the group defining the covering. We recall from our previous work \cite{bo} the results concerning the non-isotropic case and fully describe the isotropic case. We show that the necessary information to construct the Klein coverings is encoded in the 6 points on $\mathbb{P}^1$ defining the genus 2 curve. The main result of the paper is the fact that, in both cases the Prym map associated to these coverings is injective. Additionally, we provide a concrete description of the closure of the image of the Prym map inside the corresponding moduli space of polarised abelian varieties.

math.AG↗

arXiv2017

Hyperelliptic curves on $(1,4)$ polarised abelian surfaces

We investigate the number and the geometry of smooth hyperelliptic curves on a general complex abelian surface. We show that the only possibilities of genera of such curves are $2,3,4$ and $5$. We focus on the genus 5 case. We prove that up to translation, there is a unique hyperelliptic curve in the linear system of a general $(1,4)$ polarised abelian surface. Moreover, the curve is invariant with respect to a subgroup of translations isomorphic to the Klein group. We give the decomposition of the Jacobian of such a curve into abelian subvarieties displaying Jacobians of quotient curves and Prym varieties. Motivated by the construction, we prove the statement: every étale Klein covering of a hyperelliptic curve is a hyperelliptic curve, provided that the group of $2$-torsion points defining the covering is non-isotropic with respect to the Weil pairing and every element of this group can be written as a difference of two Weierstrass points.

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arXiv2016

Hyperelliptic genus 4 curves on abelian surfaces

We study smooth curves on abelian surfaces, especially for genus 4, when the complementary subvariety in the Jacobian is also a surface. We show that up to translation there is exactly one genus 4 hyperelliptic curve on a general (1, 3)-polarised abelian surface. We investigate these curves and show that their Jacobians contain a surface and its dual as complementary abelian subvarieties.

math.AG↗

arXiv2014

Non-simple principally polarised abelian varieties

The paper investigates the locus of non-simple principally polarised abelian $g$-folds. We show that the irreducible components of this locus are $\Is^g_{D}$, defined as the locus of principally polarised $g$-folds having an abelian subvariety with induced polarisation of type $D=(d_1,\ldots,d_k)$, where $k\leq\frac{g}{2}$. The main theorem produces Humbert-like equations for irreducible components of $\Is^g_{D}$ for any $g$ and $D$. Moreover, there are theorems which characterise the Jacobians of curves that are étale double covers or double covers branched in two or four points.

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