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Gavril Farkas

Publications and source records attributed to Gavril Farkas.

At least 19 recordsLinked to original sources
arXiv2026

The unirationality of $S_9^-$ and moduli spaces of pointed spin curves

We show that the moduli space of odd spin curves of genus 9 is unirational. This is the highest genus for which such a result is known. This is achieved by realizing birationally the moduli space of odd spin curves of genus g<10 as a locally trivial projective bundle over a certain (finite quotient of the) moduli space of n-pointed odd stable spin curves of genus g'<g. We then present general results on the Kodaira dimension of both components of the moduli spaces of n-pointed spin curves of genus g.

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arXiv2026

Syzygies and Koszul modules in geometry

We describe the progress in the last 10 years related to Koszul modules and syzygies of algebraic varieties. Topics discussed include the general theory of Koszul modules and resonance varieties, applications to Chen ranks of K\"ahler and hyperplane arrangement groups (Suciu's Conjecture) and connections related to syzygies of algebraic curves. Developments related to Green's Conjecture, the Secant Conjecture and the Gonality Conjecture on the resolution of line bundles on algebraic curves are also presented. Open question are proposed throughout the text.

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arXiv2025

The effective Chen ranks conjecture

Koszul modules and their associated resonance schemes are objects appearing in a variety of contexts in algebraic geometry, topology, and combinatorics. We present a proof of an effective version of the Chen ranks conjecture describing the Hilbert function of any Koszul module verifying natural conditions inspired by geometry. We give applications to hyperplane arrangements, describing in a uniform effective manner the Chen ranks of the fundamental group of the complement of every arrangement whose projective resonance is reduced. Finally, we formulate a sharp generic vanishing conjecture for Koszul modules and present a parallel between this statement and the Prym--Green Conjecture on syzygies of general Prym canonical curves.

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arXiv2025

Secant loci on moduli of Prym varieties

We present a Prym analogue of Lazarsfeld's result that curves on general polarized K3 surfaces verify the Brill-Noether Theorem, or equivalently, that their canonical embedding has no unexpected secants. We show that the Prym-canonical embedding of a curve on a general Nikulin surface (both of standard and non-standard types) has no unexpected secants. We then explain how these geometric facts suffice to determine the class of the universal difference divisor on the moduli space of stable Prym curves of (odd) genus g.

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arXiv2025

Hurwitz-Brill-Noether theory via K3 surfaces and stability conditions

We develop a novel approach to the Brill-Noether theory of curves endowed with a degree k cover of the projective line via Bridgeland stability conditions on elliptic K3 surfaces. We first develop the Brill-Noether theory on elliptic K3 surfaces via the notion of Bridgeland stability type for objects in their derived category. As a main application, we show that curves on elliptic K3 surfaces serve as the first known examples of smooth k-gonal curves which are general from the viewpoint of Hurwitz-Brill-Noether theory. In particular, we provide new proofs of the main non-existence and existence results in Hurwitz-Brill-Noether theory. Finally, using degree-k Halphen surfaces, we construct explicit examples of curves defined over number fields which are general from the perspective of Hurwitz-Brill-Noether theory.

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arXiv2025

The Kodaira dimensions of $\overline{\mathcal{M}}_{22}$ and $\overline{\mathcal{M}}_{23}$

We prove that the moduli spaces of curves of genus 22 and 23 are of general type. To do this, we calculate certain virtual divisor classes of small slope associated to linear series of rank 6 with quadric relations. We then develop new tropical methods for studying linear series and independence of quadrics and show that these virtual classes are represented by effective divisors.

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arXiv2024

Szegő kernels and Scorza quartics on the moduli space of spin curves

We describe an extension at the level of the moduli space of stable spin curves of genus g of the map associating to an ineffective spin structure its Scorza curve (equivalently, the vanishing locus of its Szegő kernel). We compute the class of the Szegő-Hodge bundle, then find an unconditional new interpretation, in terms of theta constants, of the Scorza quartic uniquely associated to an even spin structure. Our results describe the superperiod map from the moduli space of supersymmetric curves in the neighborhood of the theta-null divisor and provide a lower bound for the slope of the movable cone of the moduli space of spin curves.

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arXiv2024

Irrational components of the Hilbert scheme of points

We construct irrational irreducible components of the Hilbert scheme of points of affine n-dimensional space, for n at least 12. We start with irrational components of the Hilbert scheme of curves in P^3 and use methods developed by Jelisiejew to relate these to irreducible components of the Hilbert schemes of points of A^n. The result solves Problem XX of [J. Jelisiejew, Open problems in deformations of Artinian algebras, Hilbert schemes and around, arXiv:2307.08777, 2023].

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arXiv2024

Reduced resonance schemes and Chen ranks

The resonance varieties are cohomological invariants that are studied in a variety of topological, combinatorial, and geometric contexts. We discuss their scheme structure in a general algebraic setting and introduce various properties that ensure the reducedness of the associated projective resonance scheme. We prove an asymptotic formula for the Hilbert series of the associated Koszul module, then discuss applications to vector bundles on algebraic curves and to Chen ranks formulas for finitely generated groups, with special emphasis on Kähler and right-angled Artin groups.

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arXiv2024

Higher resonance schemes and Koszul modules of simplicial complexes

Each connected graded, graded-commutative algebra $A$ of finite type over a field $\Bbbk$ of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the (higher) Koszul modules of $A$. In this note, we investigate the geometry of the support loci of these modules, called the resonance schemes of the algebra. When $A=\Bbbk\langle Δ\rangle$ is the exterior Stanley-Reisner algebra associated to a finite simplicial complex $Δ$, we show that the resonance schemes are reduced. We also compute the Hilbert series of the Koszul modules and give bounds on the regularity and projective dimension of these graded modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group.

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arXiv2024

The minimal resolution property for points on general curves

We present an essentially complete solution to the Minimal Resolution Conjecture for general curves, determining the shape of the minimal resolution of general sets of points on a general curve C of degree d>2r-1 in P^r. Our methods also provide a proof (valid in arbitrary characteristic) of the strong version of Butler's Conjecture on the stability of syzygy bundles on a general curve of every genus at least 3, as well as of the Frobenius semistability in positive characteristic of the syzygy bundle of a general curve in the range d>2r-1.

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arXiv2023

Koszul modules with vanishing resonance in algebraic geometry

We discuss various applications of a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace in the second wedge product of a vector space. Previously Koszul modules of finite length have been used to give a proof of Green's Conjecture on syzygies of generic canonical curves. We now give applications to effective stabilization of cohomology of thickenings of algebraic varieties, divisors on moduli spaces of curves, enumerative geometry of curves on K3 surfaces and to skew-symmetric degeneracy loci. We also show that the stability of sufficiently positive rank 2 vector bundles on curves is governed by resonance.

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arXiv2023

Difference varieties and the Green-Lazarsfeld Secant Conjecture

The Green-Lazarsfeld Secant Conjecture is a generalization of Green's Conjecture on syzygies of canonical curves to the cases of arbitrary line bundles. We establish the Green-Lazarsfeld Secant Conjecture for curves of genus g in all the divisorial case, that is, when the line bundles that fail to be (p+1)-very ample form a divisor in the Jacobian of the curve.

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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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