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Sarbeswar Pal

Publications and source records attributed to Sarbeswar Pal.

11 recordsLinked to original sources
arXiv2026

On a Conjecture of Drinfeld

Let $C$ be a smooth irreducible irreducible projective curve of genus $g \ge 2$. Let $\mathcal{M}_C(n, δ)$ be the moduli space of semi-stable vector bundles on $C$ of rank $n$ and fixed determinant $δ$ of degree $d$. Then the locus of wobbly bundles is known to be closed in $\mathcal{M}_C(n, δ)$. It was announced by Laumon and attributed to Drinfeld that the wobbly locus is pure of co-dimension one, i.e., they form a divisor in $\mathcal{M}_C(n, δ)$. This is now known as Drinfeld's conjecture. In this article, we will give a proof of the conjecture when $n$ and $d$ are coprime.

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arXiv2023

Fano manifolds of Picard number one whose co-tangent bundle is algebraically completely integrable system and its endomorphisms

Let $X$ be a projective Fano manifold of Picard number one, different from the projective space. There is a folklore conjecture that any non-constant endomorphism of $X$ is an isomorphism. In the first half of this article, we will prove the folklore conjecture when the co-tangent bundle of $X$ is algebraically completely integrable system and the tangent bundle of $X$ is not nef. In the second half of the article, we will give examples of a collection of projective Fano manifolds of Picard rank one (different from the moduli space of vector bundles on algebraic curves) whose co-tangent bundles are algebraically completely integrable system. As applications of our main theorem and examples, in fact give alternative proofs of three major results appeared in three different articles.

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arXiv2022

Geometry of some moduli of bundles over a very general sextic surface for small second Chern classes and Mestrano-Simpson Conjecture

Let $S \subset \mathbb P^3$ be a very general sextic surface over complex numbers. Let $\mathcal{M}(H, c_2)$ be the moduli space of rank $2$ stable bundles on $S$ with fixed first Chern class $H$ and second Chern class $c_2$. In this article we study the configuration of points of certain reduced zero dimensional subschemes on $S$ satisfying Cayley-Bacharach property, which leads to the existence of non-trivial sections of a general memeber of the moduli space for small $c_2$. Using this study we will make an attempt to prove Mestrano-Simpson conjecture on the number of irreducible components of $\mathcal{M}(H, 11)$ and prove the conjecture partially. We will also show that $\mathcal{M}(H, c_2)$ is irreducible for $c_2 \le 10$ .

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arXiv2022

Existence of Ulrich Bundle on general Surfaces

Let $X$ be a smooth projective algebraic surface of Picard rank one with very ample canonical bundle $K_X$. We further assume that $q -1 \le χ(\mathcal{O}_X$. In this article, we will study the existence of the Ulrich bundle and its stability property of it with respect to $K_X$.

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arXiv2022

An elementary proof of Lelli Chiesa's theorem on constancy of second coordinate of gonality sequence

Let $X$ be a K3 surface and $L$ be an ample line bundle on it. In this article we will give an alternative and elementary proof of Lelli Chiesa's Theorem in the case of $r= 2$. More precisely we will prove that that under certain condition the second co-ordinate of the gonality sequence is constant along the smooth curves in the linear system $|L|$. Using Lelli Chiesa's theorem for $r \ge 3$ we also extend Lelli Chiesa's Theorem in the case of $r= 2$ in weaker condition.

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arXiv2022

Lazarsfeld-Mukai bundles on K3 surfaces associated to a pencil computing Clifford index

Let $X$ be a smooth projective K3 surface over complex numbers and $C$ be an ample curve on $X$. In this paper we will study the semistability of the Lazarsfeld-Mukai bundle $E_{C, A}$ associated to a line bundle $A$ ion $C$ such that $|A|$ is a pencil on $C$ and computes the Clifford index of $C$. We give a necessary and sufficient condition for $E_{C, A}$ being semistable.

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arXiv2018

The wobbly divisors of the moduli space of rank-$2$ vector bundles

Let $X$ be a smooth projective complex curve of genus $g \geq 2$ and let $\M_X(2,Λ)$ be the moduli space of semi-stable rank-$2$ vector bundles over $X$ with fixed determinant $Λ$. We show that the wobbly locus, i.e., the locus of semi-stable vector bundles admitting a non-zero nilpotent Higgs field is a union of divisors $\Ww_k \subset \M_X(2,Λ)$. We show that on one wobbly divisor the set of maximal subbundles is degenerate. We also compute the class of the divisors $\Ww_k$ in the Picard group of $\M_X(2,Λ)$.

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arXiv2017

Non-emptiness of Brill-Noether Loci over very general quintic hypersurface

In this article we study Brill-Noether loci of moduli space of stable bundles over smooth surfaces. We define Petri map as an analogy with the case of curves. We show the non-emptiness of certain Brill-Noether loci over very general quintic hypersurface in $\mathbb{P}^3$, and use the Petri map to produce components of expected dimension.

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arXiv2015

Semistability of Certain Bundles on Second Symmetric Power of a Curve

Let $C$ be a smooth irreducible projective curve and $E$ be a rank 2 stable vector bundle on $C$. Then one can associate a rank 4 vector bundle $\mathcal{F}_2(E)$ on $S^2(C)$, second symmetric power of $C$. Our goal in this article is to study semistability of this bundle.

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