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Ali Bajravani

Publications and source records attributed to Ali Bajravani.

11 recordsLinked to original sources
arXiv2025

Linear stability and rank two Clifford indices of algebraic curves with applications

We prove that any vector bundle computing the rank-two Clifford index of a smooth projective algebraic curve is linearly semistable. We also identify conditions under which such bundles become linearly stable, thereby addressing a question posed by A. Castorena, G. H. Hitching and E. Luna in the rank-two case. Furthermore, we demostrate that in certain special cases, this property is equivalent to the (semi)stability of the associated Lazarsfeld-Mukai bundles. This yields a positive answer, in specific cases, to a generalized version of a conjecture proposed by Mistretta and Stoppino. We also study the moduli space $S_0(n,d,5)$ of generated $α$-stable coherent systems of type $(n,d,5)$ for small values of $α$ and $n=2,3$. We show that a general element of an irreducible component of $X \subseteq S_0(2,d,5)$ or $X \subseteq S_0(3,d,5)$ is linearly stable whenever $2δ_2 \leq d \leq \frac{3g}{2}$. As an application of this, we prove that Butler's conjecture holds non-trivially for coherent systems of type $(2,d,5)$ within the given range for $d$.

math.AG↗
arXiv2024

Martens and Mumford theorems for higher rank Brill--Noether loci

Generalizing the Martens theorem for line bundles over a curve $C$, we obtain upper bounds on the dimension of the Brill--Noether locus $B^k_{n, d}$ parametrizing stable bundles of rank $n \ge 2$ and degree $d$ over $C$ with at least $k$ independent sections. This proves a conjecture of the second author and generalizes bounds obtained by him in the rank two case. We give more refined results for some values of $d$, including a generalized Mumford theorem for $n \ge 2$ when $d \le g - 1$. The statements are obtained chiefly by analysis of the tangent spaces of $B^k_{n, d}$. As an application, we show that for $n \ge 5$ the locus $B^2_{n, n(g-1)}$ is irreducible and reduced for any $C$.

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arXiv2022

Excess dimensions for Brill-Noether schemes of stable vector bundles

We extend a result by Fulton-Harris-Lazarsfeld in Brill-Noehter theory of line bundles and, as well, a result by Aprod-Sernesi in theory of Secant Loci, to the Brill-Noehter locus of stable bundles inside the moduli space of higher rank stable vector bundles on a smooth projective algebraic curve. We give some consequences of this extended result.

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arXiv2022

Excess dimensions of Brill--Noether schemes of rank two stable bundles

We use results of M. Aprodu and E. Sernesi to extend a result by Fulton--Harris--Lazarsfeld in Brill--Noether theory of line bundles %and, as well, a result by Aprod-Sernesi in theory of Secant Loci, to Brill--Noether loci of stable bundles inside the moduli space of rank two stable vector bundles on a smooth projective algebraic curve. As a consequence; if $B^k_{2, d}$ is of expected dimension, then we prove that its smoothness is equivalent with the smoothness of $B^{k}_{2, d+1}$.

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arXiv2020

Brill--Noether loci on moduli spaces of symplectic bundles over curves

The symplectic Brill--Noether locus ${\mathcal S}_{2n, K}^k$ associated to a curve $C$ parametrises stable rank $2n$ bundles over $C$ with at least $k$ sections and which carry a nondegenerate skewsymmetric bilinear form with values in the canonical bundle. This is a symmetric determinantal variety whose tangent spaces are defined by a symmetrised Petri map. We obtain upper bounds on the dimensions of various components of ${\mathcal S}_{2n, K}^k$. We show the nonemptiness of several ${\mathcal S}_{2n, K}^k$, and in most of these cases also the existence of a component which is generically smooth and of the expected dimension. As an application, for certain values of $n$ and $k$ we exhibit components of excess dimension of the standard Brill--Noether locus $B^k_{2n, 2n(g-1)}$ over any curve of genus $g \ge 122$. We obtain similar results for moduli spaces of coherent systems.

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arXiv2018

A note on the Tangent Cones of the scheme of Secant Loci

The point of this short note concerns with two facts on the scheme of secant loci. The first one is an attempt to describe the tangent cone of these schemes globally and the second one is a comparision on the dimension of the tangent spaces of various schemes of secant loci.

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arXiv2015

A note on the variety of Secant Loci

We determine non hyper elliptic curves of genus $g(C)\geq 9$, such that for some very ample line bundle on them and for some integers d and r with some prescribed assumptions, the dimension of secant loci, attains one less than its maximum value. Then we proceed to generalize and extend a problem of M. Coppens to Secant Loci.

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arXiv2015

Martens-Mumford's Theorems for Brill-Noether Schemes arising from Very Ample Line Bundles

Tangent Spaces of V^r_d(L), Specific subschemes of C_d arising from various line bundles on C, are described. Then we proceed to prove Martense Theorem for these schemes, by which we determine curves C, which for some very ample line bundle L on C and some integers r and d with d\leq h^{0}(L)-2, the subscheme V^r_d(L) might attain its maximum dimension.

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arXiv2010

Focal Varieties of Curves of Genus 6 and 8

In this paper we give a simple Torelli type theorem for curves of genus 6 and 8 by showing that these curves can be reconstructed from their Brill-Noether varieties. Among other results, it is shown that the focal variety of a general, canonical and nonhyperelliptic curve of genus 6 is a hypersurface.

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