arXiv subjects

Chandranandan Gangopadhyay

Publications and source records attributed to Chandranandan Gangopadhyay.

15 recordsLinked to original sources

arXiv2026

Equivariant principal bundles over toric Deligne--Mumford stacks

Let $\mathscr{X}(\boldsymbolΣ)$ be the toric Deligne--Mumford stack with stacky torus $\mathscr{T}$ associated to the stacky fan $\boldsymbolΣ = (N,Σ, β)$. A toric principal $H$-bundle over $\mathscr{X}(\boldsymbolΣ)$ is a principal $H$-bundle $\mathscr{P}$ equipped with a $\mathscr{T}$-action lifting the $\mathscr{T}$-action on $\mathscr{X}(\boldsymbolΣ)$, such that the $\mathscr{T}$-action and the $H$-action on $\mathscr{P}$ commute. For a connected reductive algebraic group $H$ over $\mathbb{C}$, we classify the isomorphism classes of framed toric principal $H$-bundles over $\mathscr{X}(\boldsymbolΣ)$ in terms of piecewise linear maps from $|Σ|$ to the cone over the Tits building of $H$. Using this classification, we describe the equivariant automorphism group of a toric principal $H$-bundle, and give a necessary and sufficient condition for an equivariant reduction of the structure group. We also show that every toric principal $\mathrm{GL}(r)$-bundle over the weighted projective stack $\mathbb{P}(w_0,\dots,w_n)$ splits equivariantly whenever $r < n$, and that every toric principal $H$-bundle over the weighted stacky projective line $\mathbb{P}(a,b)$ splits equivariantly when $H$ is a connected reductive algebraic group.

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arXiv2026

Birational Geometry of Quot Schemes on smooth projective curves via Stable Pairs

Let $C$ be a smooth projective curve of genus $g \geq 2$ over $\mathbb C$, and let $E^0$ be a vector bundle on $C$. We investigate the birational geometry of the Quot scheme ${\rm Quot}_C(E^0, k, n)$, which parametrizes quotients of $E^0$ of rank $k$ and degree $n$, and its fiber $\mathcal Q_L$ over ${\rm Pic}^n(C)$ for $n \gg 0$. Our main tool is the moduli space of stable pairs, which yields small $\mathbb Q$-factorial modifications (SQMs) of ${\rm Quot}_C(E^0, k, n)$ and $\mathcal Q_L$. We explicitly describe the nef, movable, and effective cones of each SQM. Consequently, we prove that $\mathcal Q_L$ is a Mori dream space and that the determinant morphism ${\rm Quot}_C(E^0, k, n) \to {\rm Pic}^n(C)$ is a Mori dream morphism.

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arXiv2026

Direct image and pullback of Parabolic vector bundles

Niels Borne established a natural correspondence between the parabolic vector bundles on curves and vector bundles on root stacks. The notions of direct image of parabolic vector bundles and pullback of parabolic vector bundles were studied in \cite{Alfaya_Biswas}. We show that these two notions correspond to the notions direct image of vector bundles on root stacks and pullback of vector bundles on root stacks respectively. Some applications of this correspondence are given.

math.AG↗

arXiv2026

Nef and Effective cones of some Quot Schemes

Let $C$ be a smooth projective curve over $\mathbb{C}$ of genus $g(C)\geqslant 3$ (respectively, $g(C)=2$). Fix integers $r,k$ such that $2\leqslant k\leqslant r-2$, (respectively, $3\leqslant k\leqslant r-2$). Let $\mathcal{Q}:={\rm Quot}_{C/\mathbb{C}}(\mathcal{O}^{\oplus r}_C, k, d)$ be the Quot scheme parametrizing rank $k$ and degree $d$ quotients of the trivial bundle of rank $r$. Let $\mathcal{Q}_L$ denote the closed subscheme of the Quot scheme parametrizing quotients such that the quotient sheaf has determinant $L$. It is known that $\mathcal{Q}_L$ is an integral, normal, local complete intersection, locally factorial scheme of Picard rank 2, when $d\gg0$. In this article we compute the nef cone, effective cone and canonical divisor of this variety when $d\gg0$. We show this variety is Fano iff $r=2k+1$.

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arXiv2025

Tautological algebra of the moduli stack of semistable bundles of rank 2 on a general curve

Our aim is to determine the tautological algebra generated by the cohomology classes of the Brill-Noether loci in the rational cohomology of the moduli stack $\mathcal{U}_C(n,d)$ of semistable bundles of rank $n$ and degree $d$. We show that for a general smooth projective curve $C$ of genus $g\geq 2$, $d=2g-2$, the tautological algebra of $ \mathcal{U}_C(2,2g-2)$ (resp. the moduli stack $\mathcal{SU}_C(2,\mathcal{L})$ of semistable bundles of rank $2$ and determinant $\mathcal{L}$ with $°(\mathcal{L})=2g-2$) is generated by the divisor classes (resp. the class of the Theta divisor $Θ$). This is previously known in rank one situation, called the (classical) Porteous formula.

math.AG↗

arXiv2025

Some recent results on the punctual Quot schemes

Let $C$ be a smooth projective curve defined over the field of complex numbers. Let $E$ be a vector bundle on $C$, and fix an integer $d\geqslant 1$. Let $\mc Q:={\rm Quot}(E,d)$ be the Quot Scheme which parameterizes all torsion quotients of $E$ of degree $d$. In this article, we survey some recent results on various invariants of $\mc Q$.

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arXiv2024

Infinitesimal deformations of some quot schemes, II

Let $C$ be an irreducible smooth complex projective curve of genus $g$, with $g_C \geqslant 2$. Let $E$ be a vector bundle on $C$ of rank $r$, with $r\geqslant 2$. Let $\mc Q:=\mc Q(E,\,d)$ be the Quot Scheme parameterizing torsion quotients of $E$ of degree $d$. We explicitly describe all deformations of $\mc Q$.

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arXiv2024

Equivariant vector bundles on complexity-one T-varieties and Bruhat-Tits buildings

We give a combinatorial classification of torus equivariant vector bundles on a (normal) projective T-variety of complexity-one. This extends the classification of equivariant line bundles on complexity-one T-varieties by Petersen-Süss on one hand, and Klyachko's classification of equivariant vector bundles on toric varieties on the other hand. A main ingredient in our classification is the classification of torus equivariant vector bundles on toric schemes over a DVR in terms of piecewise affine maps to the (extended) Bruhat-Tits building of the general linear group.

math.AG↗

arXiv2024

Picard Groups of Some Quot Schemes

Let $C$ be a smooth projective curve over the field of complex numbers $\mathbb{C}$ of genus $g(C)>0$. Let $E$ be a locally free sheaf on $C$ of rank $r$ and degree $e$. Let $\mathcal{Q}:={\rm Quot}_{C/\mathbb{C}}(E,k,d)$ denote the Quot scheme of quotients of $E$ of rank $k$ and degree $d$. For $k>0$ and $d\gg 0$ we compute the Picard group of $\mathcal{Q}$.

math.AG↗

arXiv2024

Infinitesimal deformations of some Quot schemes

Let $E$ be a vector bundle on a smooth complex projective curve $C$ of genus at least two. Let $\mathcal{Q}(E,d)$ be the Quot scheme parameterizing the torsion quotients of $E$ of degree $d$. We compute the cohomologies of the tangent bundle $T_{\mathcal{Q}(E,d)}$. In particular, the space of infinitesimal deformations of $\mathcal{Q}(E,d)$ is computed. Kempf and Fantechi computed the space of infinitesimal deformations of $\mathcal{Q}(\mathcal{O}_C,d)\,=\, C^{(d)}$. We also explicitly describe the infinitesimal deformations of $\mathcal{Q}(E,d)$.

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arXiv2023

Irreducibility of Some Nested Hilbert Schemes

Let $S$ be a smooth projective surface over $\mathbb{C}$. Let $S^{[n_1,\dots,n_k]}$ denote the nested Hilbert scheme which parametrizes zero-dimensional subschemes $ξ_{n_1} \subset \ldots \subset ξ_{n_k}$ where $ξ_i$ is a closed subscheme of $S$ of length $i$. We show that $S^{[n,m]}$, $S^{[n,m,m+1]}$, $S^{[n,n+1,m]}$, $S^{[n,n+1,m,m+1]}$, $S^{[n,n+2,m]}$ and $S^{[n,n+2,m,m+1]}$ are irreducible.

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arXiv2021

Seshadri constants on some Quot schemes

Let $E$ be a vector bundle of rank $n$ on $\mathbb{P}^1$. Fix a positive integer $d$. Let $\mathcal{Q}(E,d)$ denote the Quot scheme of torsion quotients of $E$ of degree $d$ and let $Gr(E,d)$ denote the Grassmann bundle that parametrizes the $d$-dimensional quotients of the fibers of $E$. We compute Seshadri constants of ample line bundles on $\mathcal{Q}(E,d)$ and $Gr(E,d)$.

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arXiv2020

Fundamental Group Schemes of some Quot Schemes on a smooth projective curve

Let $k$ be an algebraically closed field. Let $C$ be an irreducible smooth projective curve over $k$. Let $E$ be a locally free sheaf on $C$ of rank $\geq 2$. Fix an integer $d \geq 2$. Let $\mathcal{Q}$ denote the Quot scheme parameterizing torsion quotients of $E$ of degree $d$. In this article we compute the $S$-fundamental group scheme of $\mathcal{Q}$.

math.AG↗

arXiv2020

Nef cones of some Quot schemes on a smooth projective curve

Let $C$ be a smooth projective curve over $\mathbb C$. Let $n,d\geq 1$. Let $\mathcal Q$ be the Quot scheme parameterizing torsion quotients of the vector bundle $\mathcal O^n_C$ of degree $d$. In this article we study the nef cone of $\mathcal Q$. We give a complete description of the nef cone in the case of elliptic curves. We compute it in the case when $d=2$ and $C$ very general, in terms of the nef cone of the second symmetric product of $C$. In the case when $n\geq d$ and $C$ very general, we give upper and lower bounds for the Nef cone. In general, we give a necessary and sufficient criterion for a divisor on $\mathcal Q$ to be nef.

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arXiv2020

Automorphisms of relative Quot Schemes

Let $k$ be an algebraically closed field of characteristic zero. Let $S$ be a smooth projective variety over $k$ and let $p_S:X\rightarrow S$ be a family of smooth projective curves over $S$. Let $E$ be a vector bundle over $X$. For $s\in S$ let $X_s$ be the fibre of $p_S$ over $s$ and let $E_s$ be the restriction of $E$ to $X_s$. Fix $d\geq 1$. Let $\mathcal Q(E,d)\to S$ be the relative Quot scheme parameterizing torsion quotients of $E_s$ over $X_s$ of degree $d$ for all $s\in S$. In this article we compute the identity component of relative automorphism group scheme which parameterizes automorphisms of $\mathcal Q(E,d)$ over $S$.

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