arXiv subjects

Gavril Farkas

Publications and source records attributed to Gavril Farkas.

At least 37 records · Page 2Linked to original sources

arXiv2023

Green's conjecture for general covers

We establish Green's syzygy conjecture for classes of covers of curves of higher Clifford dimension. These curves have an infinite number of minimal pencils, in particular they do not verify a well-known Brill-Noether theoretic sufficient condition that implies Green's conjecture. Secondly, we study syzygies of curves with a fixed point free involution and prove that sections of Nikulin surfaces of minimal Picard number 9, verify the classical Green Conjecture but fail the Prym-Green Conjecture on syzygies of Prym-canonical curves. This provides an explicit locus in the moduli space R_g where Green's Conjecture is known to hold.

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arXiv2023

Du Val curves and the pointed Brill-Noether Theorem

We show that a general curve in an explicit class of what we call Du Val pointed curves satisfies the Brill-Noether Theorem for pointed curves. Furthermore, we prove that a generic pencil of Du Val pointed curves is disjoint from all Brill-Noether divisors on the universal curve. This provides explicit examples of smooth pointed curves of arbitrary genus defined over Q which are Brill-Noether general. A similar result is proved for 2-pointed curves as well using explicit curves on elliptic ruled surfaces.

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arXiv2022

Generalized De Jonquieres divisors on generic curves

The classical De Jonquieres and MacDonald formulas describe the virtual number of divisors with prescribed multiplicities in a linear system on an algebraic curve. We establish an essentially optimal result concerning the enumerative validity of these formulas in the case of a general curve of genus g.

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arXiv2022

The non-abelian Brill-Noether divisor on $\overline{\mathcal{M}}_{13}$ and the Kodaira dimension of $\overline{\mathcal{R}}_{13}$

The paper is devoted to highlighting several novel aspects of the moduli space of curves of genus 13, the first genus g where phenomena related to K3 surfaces no longer govern the birational geometry of M_g. We compute the class of the non-abelian Brill-Noether divisor on M_13 of curves that have a stable rank 2 vector bundle with many sections. This provides the first example of an effective divisor on M_g with slope less than 6+10/g. Earlier work on the Slope Conjecture suggested that such divisors may not exist. The main geometric application of our result is a proof that the Prym moduli space of genus 13 is of general type. Among other things, we also prove the Bertram-Feinberg-Mukai and the Strong Maximal Rank Conjectures on M_13

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arXiv2022

Singularities of theta divisors and the geometry of A_5

We study the codimension two locus H in A_g consisting of principally polarized abelian varieties whose theta divisor has a singularity that is not an ordinary double point. We compute the class of H in A_g for every g. For g=4, this turns out to be the locus of Jacobians with a vanishing theta-null. For g=5, via the Prym map we show that H in A_5 has two components, both unirational, which we completely describe. This gives a geometric classification of 5-dimensional ppav whose theta-divisor has a quadratic singularity of non-maximal rank. We then determine the slope of the effective cone of A_5 and show that the component N_0' of the Andreotti-Mayer divisor has minimal slope 54/7. Furthermore, the Iitaka dimension of the linear system corresponding to N_0' is submaximal.

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arXiv2022

Hodge classes on the moduli space of W(E_6)-covers and the geometry of A_6

In previous work we showed that the Hurwitz space of W(E_6)-covers of the projective line branched over 24 points dominates via the Prym-Tyurin map the moduli space A_6 of principally polarized abelian 6-folds. Here we determine the 25 Hodge classes on the Hurwitz space of W(E_6)-covers corresponding to the 25 irreducible representations of the Weyl group W(E_6). This result has direct implications to the intersection theory of the toroidal compactification A_6. In the final part of the paper, we present an alternative, elementary proof of our uniformization result on A_6 via Prym-Tyurin varieties of type W(E_6).

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arXiv2022

Linear series on general curves with prescribed incidence conditions

Using degeneration and Schubert calculus, we consider the problem of computing the number of linear series of given degree $d$ and dimension $r$ on a general curve of genus $g$ satisfying prescribed incidence conditions at $n$ points. We determine these numbers completely for linear series of arbitrary dimension when $d$ is sufficiently large, and for all $d$ when either $r=1$ or $n=r+2$. Our formulas generalize and give new proofs of recent results of Tevelev and of Cela-Pandharipande-Schmitt.

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arXiv2021

On the Kodaira dimension of Hurwitz spaces

We show that the canonical bundle of the Hurwitz stack classifying covers of genus g>1 and degree k>2 of the projective line is big. We show that all coarse moduli spaces of trigonal curves of genus g>1 are of general type.

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arXiv2021

Topological invariants of groups and Koszul modules

We provide a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace K inside the second exterior product of a vector space, as well as a sharp upper bound for its Hilbert function. This purely algebraic statement has interesting applications to the study of a number of invariants associated to finitely generated groups, such as the Alexander invariants, the Chen ranks, or the degree of growth and nilpotency class. For instance, we explicitly bound the aforementioned invariants in terms of the first Betti number for the maximal metabelian quotients of (1) the Torelli group associated to the moduli space of curves; (2) nilpotent fundamental groups of compact Kaehler manifolds; (3) the Torelli group of a free group.

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arXiv2020

Effective divisors on Hurwitz spaces

We prove the effectiveness of the canonical bundle of several Hurwitz spaces of degree k covers of the projective line from curves of genus 13<g<20.

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arXiv2020

The unirationality of the moduli space of K3 surfaces of genus 22

Using the connection discovered by Hassett between the Noether-Lefschetz moduli space of special cubic fourfolds of discriminant 42 and the moduli space F_{22} of polarized K3 surfaces of genus 22, we show that the universal K3 surface over F_{22} is unirational.

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arXiv2020

Alternating Catalan numbers and curves with triple ramification

It is known that the monodromy group of each cover of a general curve of genus g>3 equals either the symmetric or the alternating group. The classical Catalan numbers count the minimal degree covers (with symmetric monodromy) of a general curve of even genus. We solve the analogous problem for the alternating group and we determine the number of alternating covers of minimal degree 2g+1 of a general curve of genus g.

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arXiv2019

Linear syzygies on curves with prescribed gonality

We prove two statements concerning the linear strand of the minimal free resolution of a curve of fixed gonality. Firstly, we show that a general curve C of genus g of non-maximal gonality k\leq (g+1)/2 satisfies Schreyer's Conjecture, that is, b_{g-k,1}(C,K_C)=g-k, and all its highest order linear syzygies are of Eagon-Northcott type. This is a statement going beyond Green's Conjecture and predicts that all highest order linear syzygies in the canonical embedding of C are determined by the syzygies of the (k-1)-dimensional scroll containing C. Secondly, we formulate an optimal effective version of the Gonality Conjecture and prove it for general k-gonal curves. This generalizes the asymptotic Gonality Conjecture proved by Ein-Lazarsfeld and improves results of Rathmann in the case where C is a general curve of fixed gonality.

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arXiv2019

Koszul modules and Green's conjecture

We prove a strong vanishing result for finite length Koszul modules, and use it to derive Green's conjecture for every g-cuspidal rational curve over an algebraically closed field k with char(k) = 0 or char(k) >= (g+2)/2. As a consequence, we deduce that the general canonical curve of genus g satisfies Green's conjecture in this range. Our results are new in positive characteristic, whereas in characteristic zero they provide a different proof for theorems first obtained in two landmark papers by Voisin. Our strategy involves establishing two key results of independent interest: (1) we describe an explicit, characteristic-independent version of Hermite reciprocity for sl_2-representations; (2) we completely characterize, in arbitrary characteristics, the (non-)vanishing behavior of the syzygies of the tangential variety to a rational normal curve.

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arXiv2018

Quadric rank loci on moduli of curves and K3 surfaces

Given two vector bundles E and F on a variety X and a morphism from Sym^2(E) to F, we compute the cohomology class of the locus in X where the kernel of this morphism contains a quadric of prescribed rank. Our formulas have many applications to moduli theory: (i) we find a simple proof of Borcherds' result that the Hodge class on the moduli space of polarized K3 surfaces of fixed genus is of Noether-Lefschetz type, (ii) we construct an explicit canonical divisor on the Hurwitz space parametrizing degree k covers of the projective line from curves of genus 2k-1, (iii) we provide a closed formula for the Petri divisor on the moduli space of curves consisting of canonical curves which lie on a rank 3 quadric and (iv) construct myriads of effective divisors of small slope on M_g.

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arXiv2018

The resolution of paracanonical curves of odd genus

The Prym-Green conjecture predicts that the resolution of a general level p paracanonical curve of genus g is natural. Using decomposable ruled surfaces over an elliptic curve, we provide a complete solution (that is, for all levels) to this conjecture in odd genus.

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arXiv2018

The uniformization of the moduli space of principally polarized abelian 6-folds

Starting from a beautiful idea of Kanev, we construct a uniformization of the moduli space A_6 of principally polarized abelian 6-folds in terms of curves and monodromy data. We show that the general ppav of dimension 6 is a Prym-Tyurin variety corresponding to a degree 27 cover of the projective line having monodromy the Weyl group of the E_6 lattice. Along the way, we establish numerous facts concerning the geometry of the Hurwitz space of such E_6-covers, including: (1) a proof that the canonical class of the Hurwitz space is big, (2) a concrete geometric description of the Hodge-Hurwitz eigenbundles with respect to the Kanev correspondence and (3) a description of the ramification divisor of the Prym-Tyurin map from the Hurwitz space to A_6 in the terms of syzygies of the Abel-Prym-Tyurin curve.

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arXiv2017

Syzygies of Prym and paracanonical curves of genus 8

By analogy with Green's Conjecture on syzygies of canonical curves, the Prym-Green conjecture predicts that the resolution of a general level p paracanonical curve of genus g is natural. The Prym-Green Conjecture is known to hold in odd genus for almost all levels. Probabilistic arguments strongly suggested that the conjecture might fail for level 2 and genus 8 or 16. In this paper, we present three geometric proofs of the surprising failure of the Prym-Green Conjecture in genus 8, hoping that the methods introduced here will shed light on all the exceptions to the Prym-Green Conjecture for genera with high divisibility by 2.

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