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Praveen Kumar Roy

Publications and source records attributed to Praveen Kumar Roy.

10 recordsLinked to original sources
arXiv2025

Positivity on simple $G$-varieties

Let $X$ be a normal projective variety equipped with an action of a semisimple algebraic group $G$, and assume that $X$ contains a unique closed orbit. Let $B$ be a Borel subgroup of $G$ and let $E$ be a $B$-equivariant vector bundle on $X$. In this article, we prove that $E$ is ample (respectively, nef) if and only if its restriction to the finite set of $B$-stable curves in $X$ is ample (respectively, nef). Moreover, we compute the nef cone of the blow-up of a nonsingular simple $G$-projective variety $X$ at a unique $B$-fixed point $x^-$, referred to as the sink of $X$. As an application, when $X$ is nonsingular, we calculate the Seshadri constants of any ample line bundle (not necessarily $G$-equivariant) at $x^-$. In addition, we compute the Seshadri constants of $B$-equivariant vector bundles at $x^{-}$.

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arXiv2025

Positivity on Blow-up of hyperelliptic surfaces

Let $X_r$ denote the blow-up of the hyperelliptic surface $X$ at $r$ very general points. In this paper, we first provide a criterion for the ampleness of a line bundle on $X_r$ and compare it with an existing result. We then study the multi-point Seshadri constants of ample line bundles on hyperelliptic surfaces $X$. Next, we compute single-point Seshadri constants on $X_r$ for specific ample line bundles on odd types. Furthermore, we show that the global Seshadri constants for certain ample line bundles on blow-up of hyperelliptic surfaces are rational.

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arXiv2025

Exploring the interplay of semistable vector bundles and their restrictions on reducible curves

Let $C$ be a comb-like curve over $\mathbb{C}$, and $E$ be a vector bundle of rank $n$ on $C$. In this paper, we investigate the criteria for the semistability of the restriction of $E$ onto the components of $C$ when $E$ is given to be semistable with respect to a polarization $w$. As an application, assuming each irreducible component of $C$ is general in its moduli space, we investigate the $w$-semistability of kernel bundles on such curves, extending the results (completely for rank two and partially for higher rank) known in the case of a reducible nodal curve with two smooth components, but here, using different techniques.

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arXiv2024

The fundamental group of Galois covers of surfaces with octahedral envelope

We compute the fundamental group of the Galois cover of a surface of degree~$8$, with singularities of degree $4$, whose degeneration envelope is isomorphic to an octahedron. The group is shown to be a metabelian group of order $2^{23}$. The computation amalgamates local groups, classified elsewhere, by an iterative combination of computational and group theoretic methods. Three simplified surfaces, for which the fundamental group of the Galois cover is trivial, demonstrate how nontrivial cycles in the degenerated surface complicate the computation.

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arXiv2024

Seshadri constants of curve configurations on surfaces

Let $X$ be a complex nonsingular projective surface and let $L$ be an ample line bundle on $X$. We study multi-point Seshadri constants of $L$ at singular points of certain arrangements of curves on $X$. We pose some questions about such Seshadri constants and prove some results in the case of star arrangements of curves. We also study the configurational Seshadri constants for curve arrangements on surfaces and compare them with the usual Seshadri constants. We give several examples illustrating the properties that we study.

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arXiv2023

Seshadri constants on some blow-ups of projective spaces

Let $X^n_{r,s}$ denote the blow-up of $\mathbb{P}^n$ along $r$ general lines and $s$ general points. In this paper, we focus on $l$-very ample line bundles on $X^n_{0,s}$ and investigate their Seshadri constants with some restrictions on $s$. Additionally, we compute the nef cone of $X^3_{r,0}$ for $r\leq 6$ and study the Seshadri constants of some ample line bundles on it. We also examine the Seshadri constants of some ample line bundles on $X^4_{r,0}$ ($r\leq 7$) and $X^5_{r,0}$ ($r \leq 5$).

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arXiv2023

Fundamental groups of highly symmetrical curves and Fermat line arrangments

We showcase a computation of the fundamental group of $\mathbb{CP}^2 - \mathcal{C}$ when $\mathcal{C}$ is a curve admitting a lot of symmetries. In particular, let $\mathcal{C}$ denote the Fermat line arrangement in $\mathbb{CP}^2$ defined by the vanishing locus of homogeneous polynomial $(x^n-y^n)(y^n-z^n)(z^n-x^n)$. In this article, we compute the fundamental group $π_1(\mathbb{CP}^2-\mathcal{C})$ of complement of this line arrangement in the complex projective plane. We show that this group is semi-direct product of $G$ and $F_n$, i.e., $π_1(\mathbb{CP}^2-\mathcal{C}, \overlineε) = G \rtimes F_{n}$, where $G$ and $F_n$ is defined in 4.3, and 1.2 respectively.

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arXiv2019

Some Results on Seshadri constants on Surfaces of general type

We prove two new results for Seshadri constants on surfaces of general type. Let $X$ be a surface of general type. In the first part, inspired by \cite{B-S}, we list the possible values for the multi-point Seshadri constant $\varepsilon(K_X,x_1,x_2,...,x_r)$ when it lies between $0$ and $1/r$, where $K_X$ is the canonical line bundle on $X$. In the second part, we assume $X$ of the form $C \times C$, where $C$ is a general smooth curve of genus $g \geq 2$. Given such $X$ and an ample line bundle $L$ on $X$ with some conditions on it, we show that the global Seshadri constant of $L$ is a rational number.

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arXiv2018

Seshadri constants on hyperelliptic surfaces

We prove new results on single point Seshadri constants for ample line bundles on hyperelliptic surfaces. Given a hyperelliptic surface $X$ and an ample line bundle $L$ on $X$, we show that the least Seshadri constant $\varepsilon(L)$ of $L$ is a rational number when $X$ is not of type 6. We also prove new lower bounds for the Seshadri constant $\varepsilon(L,1)$ of $L$ at a very general point.

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