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Vassil Kanev

Publications and source records attributed to Vassil Kanev.

11 recordsLinked to original sources
arXiv2025

Hurwitz moduli varieties parameterizing pointed covers of an algebraic curve with a fixed monodromy group

Given a smooth, projective curve $Y$, a point $y_0 \in Y$, a positive integer $n$, and a transitive subgroup $G$ of the symmetric group $S_{d}$ we study smooth, proper families, parameterized by algebraic varieties, of pointed degree $d$ covers of $(Y,y_0)$, $(X,x_{0})\to (Y,y_0)$, branched in $n$ points of $Y\setminus y_{0}$, whose monodromy group equals $G$. We construct a Hurwitz space $H$, an algebraic variety whose points are in bijective correspondence with the equivalence classes of pointed covers of $(Y,y_0)$ of this type. We construct explicitly a family parameterized by $H$, whose fibers belong to the corresponding equivalence classes, and prove that it is universal. We use classical tools of algebraic topology and of complex algebraic geometry.

math.AG↗
arXiv2024

Hurwitz moduli varieties parameterizing Galois covers of an algebraic curve

Given a smooth, projective curve $Y$, a finite group $G$ and a positive integer $n$ we study smooth, proper families $X\to Y\times S\to S$ of Galois covers of $Y$ with Galois group isomorphic to $G$ branched in $n$ points, parameterized by algebraic varieties $S$. When $G$ is with trivial center we prove that the Hurwitz space $H^G_n(Y)$ is a fine moduli variety for this moduli problem and construct explicitly the universal family. For arbitrary $G$ we prove that $H^G_n(Y)$ is a coarse moduli variety. For families of pointed Galois covers of $(Y,y_0)$ we prove that the Hurwitz space $H^G_n(Y,y_0)$ is a fine moduli variety, and construct explicitly the universal family, for arbitrary group $G$. We use classical tools of algebraic topology and of complex algebraic geometry.

math.AG↗
arXiv2016

Unirationality of Hurwitz spaces of coverings of degree <= 5

Let $Y$ be a smooth, projective curve of genus $g\geq 1$ over the complex numbers. Let $H^0_{d,A}(Y)$ be the Hurwitz space which parametrizes coverings $p:X \to Y$ of degree $d$, simply branched in $n=2e$ points, with monodromy group equal to $S_d$, and $det(p_{*}O_X/O_Y)$ isomorphic to a fixed line bundle $A^{-1}$ of degree $-e$. We prove that, when $d=3, 4$ or $5$ and $n$ is sufficiently large (precise bounds are given), these Hurwitz spaces are unirational. If in addition $(e,2)=1$ (when $d=3$), $(e,6)=1$ (when $d=4$) and $(e,10)=1$ (when $d=5$), then these Hurwitz spaces are rational.

math.AG↗
arXiv2007

Polarization types of isogenous Prym-Tyurin varieties

Let p:C-->Y be a covering of smooth, projective curves which is a composition of π:C-->C' of degree 2 and g:C'-->Y of degree n. Let f:X-->Y be the covering of degree 2^n, where the curve X parametrizes the liftings in C^{(n)} of the fibers of g:C'-->Y. Let P(X,δ) be the associated Prym-Tyurin variety, known to be isogenous to the Prym variety P(C,C'). Most of the results in the paper focus on calculating the polarization type of the restriction of the canonical polarization of JX on P(X,δ). We obtain the polarization type when n=3. When Y=P^1 we conjecture that P(X,δ) is isomorphic to the dual of the Prym variety P(C,C'). This was known when n=2, we prove it when n=3, and for arbitrary n if π:C-->C' is étale. Similar results are obtained for some other types of coverings.

math.AG↗
arXiv2005

Irreducibility of Hurwitz spaces

Graber, Harris and Starr proved, when n >= 2d, the irreducibility of the Hurwitz space H^0_{d,n}(Y) which parametrizes degree d coverings of a smooth, projective curve Y of positive genus, simply branched in n points, with full monodromy group S_d (math.AG/0205056). We sharpen this result and prove that H^0_{d,n}(Y) is irreducible if n >= max{2,2d-4} and in the case of elliptic Y if n >= max{2,2d-6}. We extend the result to coverings simply branched in all but one point of the discriminant. Fixing the ramification multiplicities over the special point we prove that the corresponding Hurwitz space is irreducible if the number of simply branched points is >= 2d-2. We study also simply branched coverings with monodromy group different from S_d and when n is large enough determine the corresponding connected components of H_{d,n}(Y). Our results are based on explicit calculation of the braid moves associated with the standard generators of the n-strand braid group of Y.

math.AG↗
arXiv2003

Hurwitz spaces of triple coverings of elliptic curves and moduli spaces of abelian threefolds

We prove that the moduli spaces A_3(D) of polarized abelian threefolds with polarizations of types D=(1,1,2), (1,2,2), (1,1,3) or (1,3,3) are unirational. The result is based on the study of families of simple coverings of elliptic curves of degree 2 or 3 and on the study of the corresponding period mappings associated with holomorphic differentials with trace 0. In particular we prove the unirationality of the Hurwitz space H_{3,A}(Y) which parameterizes simply branched triple coverings of an elliptic curve Y with determinants of the Tschirnhausen modules isomorphic to A^{-1}.

math.AG↗
arXiv2003

Hurwitz spaces of quadruple coverings of elliptic curves and the moduli space of abelian threefolds A_3(1,1,4)

We prove that the moduli space A_3(1,1,4) of polarized abelian threefolds with polarization of type (1,1,4) is unirational. By a result of Birkenhake and Lange this implies the unirationality of the isomorphic moduli space A_3(1,4,4). The result is based on the study the Hurwitz space H_{4,n}(Y) of quadruple coverings of an elliptic curve Y simply branched in n points. We prove the unirationality of its codimension one subvariety H^{0}_{4,A}(Y) which parametrizes quadruple coverings π:X --> Y with Tschirnhausen modules isomorphic to A^{-1}, where A\in Pic^{n/2}Y, and for which π^*:J(Y)--> J(X) is injective. This is an analog of the result of Arbarello and Cornalba that the Hurwitz space H_{4,n}(P^1) is unirational.

math.AG↗
arXiv1998

Chordal varieties of Veronese varieties and catalecticant matrices

It is proved that the chordal variety of the Veronese variety v_d(P^n) is projectively normal, arithmetically Cohen-Macaulay and its homogeneous ideal is generated by the 3 x 3 minors of two catalecticant matrices. These results are generalized to the catalecticant varieties Gor_{\leq}(T) with t_1 = 2. We also give a simplified proof of a theorem of O. Porras about the rank varieties of symmetric tensors.

math.AG↗
arXiv1993

Recovering of curves with involution by extended Prym data

With every smooth, projective algebraic curve $\tilde{C}$ with involution $σ:\tilde{C}\longrightarrow \tilde{C}$ without fixed points is associated the Prym data which consists of the Prym variety $P:=(1-σ)J(\tilde{C})$ with principal polarization $Ξ$ such that $2Ξ$ is algebraically equivalent to the restriction on $P$ of the canonical polarization $Θ$ of the Jacobian $J(\tilde{C})$. In contrast to the classical Torelli theorem the Prym data does not always determine uniquely the pair $(\tilde{C},σ)$ up to isomorphism. In this paper we introduce an extension of the Prym data as follows. We consider all symmetric theta divisors $Θ$ of $J(\tilde{C})$ which have even multiplicity at every point of order 2 of $P$. It turns out that they form three $P_2$ orbits. The restrictions on $P$ of the divisors of one of the orbits form the orbit $\{ 2Ξ\} $, where $Ξ$ are the symmetric theta divisors of $P$. The other restrictions form two $P_2$-orbits $O_1,O_2\subset \mid 2Ξ\mid $. The extended Prym data consists of $(P,Ξ)$ together with $O_1,O_2$. We prove that it determines uniquely the pair $(\tilde{C} ,σ)$ up to isomorphism provided $g(\tilde{C})\geq 3$. The proof is analogous to Andreotti's proof of Torelli's theorem and uses the Gauss map for the divisors of $O_1,O_2$. The result is an analog in genus $>1$ of a classical theorem for elliptic curves.

alg-geom↗