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Montserrat Teixidor i Bigas

Publications and source records attributed to Montserrat Teixidor i Bigas.

At least 19 recordsLinked to original sources
arXiv2026

Divisors and harmonic morphisms on metric graphs of pseudocompact type

A metric graph is of pseudocompact type if identifying parallel edges produces a tree. We give a constructive proof that, for such graphs, divisorial $d$-gonality is equivalent to the existence of a degree $d$ harmonic morphism to a tree. This mirrors the algebraic correspondence, for curves of compact type, between limit linear series of dimension one and admissible covers. We also deduce lifting results for positive-rank divisors, with a genus-preserving refinement when identifying parallel edges produces a path. Finally, we study the Brill-Noether theory in the path case.

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arXiv2026

Some reducible and irreducible Brill-Noether loci

We investigate limit linear series on chains of elliptic curves, giving a simple proof of a conjecture of Farkas stating the existence of curves with a theta-characteristic with a given number of sections for the expected range of genera. Using the additional structure afforded by considering limit linear series on chains of elliptic curves, we find examples of reducible Brill--Noether loci, admitting at least two components, with and without a theta-characteristic respectively. This allows us to display reducible Hilbert schemes for $r\ge 3$ and $d=g-1$. We also give examples of Brill--Noether loci with three components. On the positive side, we provide optimal bounds on the degree under which Brill--Noether loci are irreducible when $r=2$.

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arXiv2024

Brill-Noether loci

Brill-Noether loci ${\mathcal M}^r_{g,d}$ are those subsets of the moduli space ${\mathcal M}_g$ determined by the existence of a linear series of degree $d$ and dimension $r$. By looking at non-singular curves in a neighborhood of a special chain of elliptic curves, we provide a new proof of the non-emptiness of the Brill-Noether loci when the expected codimension satisfies $-g+r+1\le ρ(g,r,d)\le 0$ and prove that for a generic point of a component of this locus, the Petri map is onto. As an application, we show that Brill-Noether loci of the same codimension are distinct when the codimension is not too large, substantially generalizing the known result in codimensions 1 and 2. We also provide a new technique for checking that Brill-Noether loci are not included in each other.

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arXiv2023

Ranks of maps of vector bundles

We generalize to vector bundles the techniques introduced for line bundles in prior work of the author with Liu, Osserman and Zhang. We then use this method to prove the injectivity of the Petri map for vector bundles and the surjectivity of a map related to deformation theory of Poincaré sheaves.

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arXiv2022

Successive extensions of vector bundles on curves

We show that on a generic curve, a bundle obtained by successive extensions is stable. We compute the dimension of the set of such extensions. We use degeneration methods specializing the curve to a chain of elliptic components

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arXiv2021

Brill-Noether loci with ramification at two points

We prove the injectivity of the Petri map for linear series on a general curve with given ramification at two generic points. We also describe the components of such a set of linear series on a chain of elliptic curves.

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arXiv2020

Rational Curves on Moduli Spaces of Vector Bundles

We completely describe the components of expected dimension of the Hilbert Scheme of rational curves of fixed degree $k$ in the moduli space ${\rm SU}_{C}(r,L)$ of semistable vector bundles of rank $r$ and determinant $L$ on a curve $C$. We show that for every $k \geq 1$ there are ${\rm gcd}(r, °L)$ unobstructed components. In addition, if $k$ is divisible by $r_1(r-r_1)(g-1)$ for $1\le r_1\le r-1$, there is an additional obstructed component of the expected dimension for each such $r_1$. We construct families of obstructed components and show that their generic point is not the generic vector bundle of given rank and determinant. Finally, we also obtain an upper bound on the degree of rational connectedness of ${\rm SU}_{C}(r,L)$ which is linear in the dimension.

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arXiv2020

Coherent systems on the projective line

It is well known that there are no stable bundles of rank greater than 1 on the projective line. In this paper, our main purpose is to study the existence problem for stable coherent systems on the projective line when the number of sections is larger than the rank. We include a review of known results, mostly for a small number of sections.

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arXiv2018

The strong maximal rank conjecture and moduli spaces of curves

Building on recent work of the authors, we use degenerations to chains of elliptic curves to prove two cases of the Aprodu-Farkas strong maximal rank conjecture, in genus $22$ and $23$. This constitutes a major step forward in Farkas' program to prove that the moduli spaces of curves of genus $22$ and $23$ are of general type. Our techniques involve a combination of the Eisenbud-Harris theory of limit linear series, and the notion of linked linear series developed by the second author.

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arXiv2017

Limit linear series and tropical divisors on chains

We describe the space of Eisenbud-Harris limit linear series on a chain of elliptic curves and compare it with the theory of divisors on tropical chains. Either model allows to compute some invariants of Brill-Noether theory using combinatorial methods. We introduce effective limit linear series.

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arXiv2016

Genera of Brill-Noether curves and staircase paths in Young tableaux

In this paper, we compute the genus of the variety of linear series of rank $r$ and degree $d$ on a general curve of genus $g$, with ramification at least $α$ and $β$ at two given points, when that variety is 1-dimensional. Our proof uses degenerations and limit linear series along with an analysis of random staircase paths in Young tableaux, and produces an explicit scheme-theoretic description of the limit linear series of fixed rank and degree on a generic chain of elliptic curves when that scheme is itself a curve.

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arXiv2014

Invariants of the Brill-Noether curve

For a projective nonsingular curve of genus $g$, the Brill-Noether locus $W^r_d(C)$ parametrizes line bundles of degree $d$ over $C$ with at least $r+1$ sections. When the curve is generic and the Brill-Noether number $ρ(g,r,d)$ equals $1$, one can then talk of the \emph{Brill-Noether curve}. In this paper, we explore the first two invariants of this curve, giving a new way of calculating the genus of this curve and computing its gonality when $C$ has genus 5.

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arXiv2014

Linked symplectic forms and limit linear series in rank 2 with special determinant

We generalize the prior linked symplectic Grassmannian construction, applying it to to prove smoothing results for rank-2 limit linear series with fixed special determinant on chains of curves. We apply this general machinery to prove new results on nonemptiness and dimension of rank-2 Brill-Noether loci in a range of degrees.

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arXiv2012

Some examples of vector bundles in the base locus of the generalized theta divisor

This paper shows that on the moduli space of semi-stable vector bundles of fixed rank and determinant (of any degree) on a smooth curve of genus at least two, the base locus of the generalized theta divisor is large provided the rank is sufficiently large. It also shows that the base locus is large for positive multiples of the theta divisor. This work extends results already known for the case where the determinant is of degree zero.

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arXiv2012

Singularities of Brill-Noether loci for vector bundles on a curve

In this paper we consider the singularities of the varieties parameterizing stable vector bundles of fixed rank and degree with sections on a smooth curve of genus at least two. In particular, we extend results of Y. Laszlo, and of the second author, regarding the singularities of generalized theta divisors.

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arXiv2012

Petri map for vector bundles near good bundles

We show that on a generic curve and under some conditions on the degree and genus, there exists a component B of the locus of stable vector bundles of rank r and degree d with at least k sections of the expected dimension such that for a generic vector bundle E of B, the Petri map is injective.

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arXiv2011

Linked alternating forms and linked symplectic Grassmannians

Motivated by applications to higher-rank Brill-Noether theory and the Bertram-Feinberg-Mukai conjecture, we introduce the concepts of linked alternating and linked symplectic forms on a chain of vector bundles, and show that the linked symplectic Grassmannians parametrizing chains of subbundles isotropic for a given linked symplectic form has good dimensional behavior analogous to that of the classical symplectic Grassmannian.

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