arXiv subjects

Nabanita Ray

Publications and source records attributed to Nabanita Ray.

12 recordsLinked to original sources

arXiv2026

Seshadri constants of Higgs Vector bundles

We define Seshadri constants for Higgs bundles on smooth projective varieties over algebraically closed fields of characteristic zero. This definition is inspired by and analogous to the notion of Seshadri constants for ordinary vector bundles. We prove a series of properties of Higgs Seshadri constants which are analogous to the corresponding properties in the case of ordinary Seshadri constants. In particular, we prove a Seshadri criterion for Higgs ampleness and prove that Higgs Seshadri constants can be computed by restriction to curves.

math.AG↗

arXiv2026

Positivity of Higgs Vector Bundles

In \cite{BCO25}, Bruzzo, Capasso and Otero extended the notion of ampleness of vector bundles to the more general context of Higgs bundles. But the ampleness of Higgs bundles did not coincide with the ampleness of vector bundles when the Higgs field is zero. We modify the definition of ample Higgs bundles that results in removal of this discrepancy. Invoking this definition, we study various properties of Higgs ample vector bundles. In particular, we prove a Barton-Kleimann type criterion to characterize the Higgs ample vector bundles.

math.AG↗

arXiv2025

Positive cones of $b$-divisor classes

In this article, we define the notion of ample Cartier $b$-divisor classes by using the notion of Seshadri constants for Cartier $b$-divisor classes. In particular, we have shown that the set of all ample Cartier $b$-divisor classes forms a convex cone inside the nef cone of Cartier $b$-divisor classes. Furthermore, we have studied various properties of these Cartier ample $b$-divisor classes. We have also given an equivalent characterization of big Cartier $b$-divisor classes in terms of volume function of the pseudo-effective Cartier $b$-divisor classes. More specifically, we prove that the set of all big Cartier $b$-divisor classes form a convex cone. Finally we have investigated how the nef Cartier $b$-divisor classes behave under the pullback.

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arXiv2024

On Weak bounded negativity conjecture

In the first part of this article, we give bounds on self-intersections $C^2$ of integral curves $C$ on blow-ups $Bl_nX$ of surfaces $X$ with the anti-cannonical divisor $-K_X$ effective. In the last part, we prove the weak bounded negativity for self-intersections $C^2$ of integral curves $C$ in a family of surfaces $f:Y\longrightarrow B$ where $B$ is a smooth curve.

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arXiv2024

On Stability of Syzygy Bundles

In this article, we investigate the stability of syzygy bundles corresponding to ample and globally generated vector bundles on smooth irreducible projective surfaces.

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arXiv2023

Positivity and base loci for vector bundles revisited

We give equivalent descriptions for the augmented and diminished base loci of vector bundles in characteristic zero. We show that these base loci behave well under pullback, tensor product, and direct sum. Pathological behavior is observed on some nonsplit exact sequences.

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arXiv2022

Slope Semistability and Positive cones of Grassmann bundles

Let $E$ be a vector bundle of rank $r$ on a smooth complex projective variety $X$. In this article, we compute the nef and pseudoeffective cones of divisors in the Grassmann bundle $Gr_X(k,E)$ parametrizing $k$-dimensional subspaces of the fibers of $E$, where $1\leq k \leq rank(E)$, under assumptions on $X$ as well as on the vector bundle $E$. In particular, we show that nef cone and the pseudoeffective cone of $Gr_X(k,E)$ coincide if and only if $E$ is a slope semistable bundle on $X$ with $c_2(End(E))=0$. We also discuss about the nefness and ampleness of the universal quotient bundle $Q_k$ on $Gr_X(k,E)$.

math.AG↗

arXiv2022

Seshadri constants of parabolic vector bundles

Let $X$ be a complex projective variety, and let $E_{\ast}$ be a parabolic vector bundle on $X$. We introduce the notion of \textit{parabolic Seshadri constants} of $E_{\ast}$. It is shown that these constants are analogous to the classical Seshadri constants of vector bundles, in particular, they have parallel definitions and properties. We prove a Seshadri criterion for parabolic ampleness of $E_{\ast}$ in terms of parabolic Seshadri constants. We also compute parabolic Seshadri constants for symmetric powers and tensor products of parabolic vector bundles.

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arXiv2020

Stability and semi-stability of (2,2)-type surfaces

We describe the GIT compactification of the moduli of (2,2)-type effective divisors of $\mathbb{P}^1\times\mathbb{P}^2$ (i.e., surfaces of the linear system $\vert \pi_1^*\mathcal{O}_{\mathbb{P}^1}(2)\otimes \pi_2^*\mathcal{O}_{\mathbb{P}^2}(2)\vert$ ) which are generically Del Pezzo surfaces of degree two. In order to get the compactification, we characterize stable and semi-stable (2,2)-type surfaces, and also determine the equivalence classes of strictly semi-stable (2,2)-type surfaces. Moreover, we describe the boundary of the moduli of (2,2)-type surfaces.

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arXiv2019

Weyl and Zariski chambers on projective surfaces

Let $X$ be a nonsingular complex projective surface. The Weyl and Zariski chambers give two interesting decompositions of the big cone of $X$. We study these two decompositions and determine when a Weyl chamber is contained in the interior of a Zariski chamber and vice versa. We also determine when a Weyl chamber can intersect non-trivially with a Zariski chamber.

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arXiv2019

Nef cones of projective bundles over surfaces and Seshadri constants

In this article, we give a description of the closed cone of curves of the projective bundle $\mathbb{P}(E)$ over a smooth projective variety $X$. Using duality, we then calculate the nef cone of divisors in $\mathbb{P}(E)$ over some special surfaces $X$ and for some special bundles on $X$. As an application, we also calculate the Seshadri constants of semistable ample vector bundles with vanishing discriminant on some special ruled surfaces at special points.

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arXiv2018

Geometry of $\mathbb{P}^{2}$ blown up at seven points

In this paper, we prove that $\mathbb{P}^2$ blown up at seven general points admits a conic bundle structure over $\mathbb{P}^1$ and it can be embedded as $(2,2)$ divisor in $\mathbb{P}^{1}\times\mathbb{P}^{2}$. Conversely, any smooth surface in the complete linear system $\mid (2,2) \mid$ of $\mathbb{P}^{1}\times\mathbb{P}^{2}$ is obtained by blowing up $\mathbb{P}^2$ at seven points. We also show any smooth surface linearly equivalent to $(2,2)$ in $\mathbb{P}^{1}\times\mathbb{P}^{2}$ has at most four $(-2)$ curves (the curve which has self intersection $(-2)$).

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