arXiv subjects

Jagadish Pine

Publications and source records attributed to Jagadish Pine.

6 recordsLinked to original sources

arXiv2026

On the triviality of direct image of coherent sheaves

Let $\pi\,:\, X \,\longrightarrow\, Y$ be a finite morphism of projective varieties defined over an algebraically closed field of characteristic zero. We study the necessary and sufficient criteria for $\pi$ such that there exists a coherent sheaf $E$ on $X$ whose direct image $\pi_*E$ is a trivial vector bundle on $Y$ of positive rank. When $X$ is smooth, and $Y$ is Cohen-Macaulay, such a coherent sheaf is necessarily locally free. We show that the existence of such a coherent sheaf $E$ is guided by the properties of the branching divisor of $\pi$. When the covering $\pi\,:\, X \,\longrightarrow\, Y$ is admissible abelian Galois, we give a complete answer. As an application, it is shown that every smooth admissible abelian Galois covering of $\mathbb{P}^n$ supports an Ulrich bundle.

math.AG↗

arXiv2024

A remark on a theorem of Narasimhan and Ramanan

In this short note, we provide an alternative proof of a notable theorem by Narasimhan and Ramanan. The theorem states that the moduli space of $S$-equivalence classes of semistable rank $2$ vector bundles over a curve $X$ of genus $2$ with trivial determinant is isomorphic to $\mathbb{P}^3$. Our proof relies on a criterion by Bauer and Szemberg, which characterizes projective spaces among smooth Fano varieties using Seshadri constants.

math.AG↗

arXiv2024

Ulrich bundles on cyclic coverings of projective spaces

We prove the existence of Ulrich bundles on cyclic coverings of $\mathbb{P}^n$ of arbitrary degree $d$. Given a relatively Ulrich bundle on a complete intersection subvariety, we construct a relatively Ulrich bundle on the ambient variety. As an application, we prove that there exists a rank $d$ Ulrich bundle on a generic cyclic covering of $\mathbb{P}^{2}$ of degree $d$, provided that the degree $d \cdot k$ of the branch divisor is even. When $d \cdot k$ is odd, we also provide an estimation of the rank of the Ulrich bundle on a generic cyclic covering of $\mathbb{P}^2$.

math.AG↗

arXiv2024

$\ell$-away ACM line bundles on a nonsingular cubic surface

Let $X \subset \mathbb P^3$ be a nonsingular cubic hypersurface. Faenzi (\cite{F}) and later Pons-Llopis and Tonini (\cite{PLT}) have completely characterized ACM line bundles over $X$. As a natural continuation of their study in the non-ACM direction, in this paper, we completely classify $\ell$-away ACM line bundles (introduced recently by Gawron and Genc (\cite{GG})) over $X$, when $\ell \leq 2$. For $\ell\geq 3$, we give examples of $\ell$-away ACM line bundles on $X$ and for each $\ell \geq 1$, we establish the existence of smooth hypersurfaces $X^{(d)}$ of degree $d >\ell$ in $\mathbb P^3$ admitting $\ell$-away ACM line bundles.

math.AG↗

arXiv2024

Seshadri constants on some flag bundles

Let $X$ be a smooth complex projective curve and let $E$ be a vector bundle on $X$ which is not semistable. We consider a flag bundle $π: \text{Fl}(E) \to X$ parametrizing certain flags of fibers of $E$. The dimensions of the successive quotients of the flags are determined by the ranks of vector bundles appearing in the Harder-Narasimhan filtration of $E$. We compute the Seshadri constants of nef line bundles on $\text{Fl}(E)$.

math.AG↗

arXiv2022

Universal parabolic moduli over $\overline{M}_{_{g, n}}$

In this article we will construct a universal moduli space of stable parabolic vector bundles over the moduli space of marked Deligne-Mumford stable curves $\overline{M}_{_{g, n}}$. The objects that appear over the boundary of $\overline{M}_{_{g, n}}$ i.e., over singular curves will remain vector bundles. The total space and the fibers over $\overline{M}_{_{g, n}}$ will have good singularities.

math.AG↗