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Usha N. Bhosle

Publications and source records attributed to Usha N. Bhosle.

9 recordsLinked to original sources
arXiv2026

Picard bundles and Twisted Picard bundles on the Jacobian of a curve

Let $Y$ denote an irreducible projective curve with at most nodes as singularities and defined over an algebraically closed field of characteristic zero. We study the restriction of the twisted Picard bundles on the compactified Jacobian $\overline{J}(Y)$ of $Y$ to the embedded curve in $\overline{J}(Y)$. As an application, we show that for $g =2$ and each integer $r \ge 3$, there is a two-dimensional family of stable ACM bundles on the compactified Jacobian which has the Picard bundle in its limit. We define an embedding $α_Y$ of the (generalised) Jacobian $J(Y)$ in the moduli space $U^s_Y(n,d)$ of stable vector bundles on $Y$ using a twisted restriction $E_Y$ of a Picard bundle to $Y \subset J(Y)$. We show that (under suitable conditions) the restriction of the universal bundle $\mathcal{U}$ to $Y \times J(Y)$ is stable for suitable polarisation. For the embedding of a smooth curve $Y$ given by $E_Y \otimes B, B$ a line bundle of degree $b$, we show that the restriction of the Picard bundle on $U^s_Y(n, d+nb)$ to $J(Y)$ is $θ$-semistable for $b \ge 2g-1$ and $θ$-stable for $b \ge 2g$. We also determine the relation between the restriction of the theta divisor on $U^s_Y(n,d+nb)$ to $J(Y)$ and the theta divisor $θ$ on $\overline{J}(Y)$.

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arXiv2026

Hitchin maps and parabolic Hitchin maps on the moduli spaces of Hitchin sheaves on nodal curves

We study the Hitchin maps on the moduli spaces of Hitchin sheaves and parabolic Hitchin sheaves on a nodal integral curve $Y$. We study their fibres, the BNR correspondences and the relation of the restriction of the Hitchin map with very stable bundles. As an application, in the "coprime" case we prove that the parabolic Poincaré sheaf on $U_{par}(n, ξ)\times Y$ is parabolic stable where $U_{par}(n, ξ)$ is a fixed determinant moduli space of parabolic sheaves on $Y$.

math.AG↗
arXiv2020

Poincaré bundle for the fixed determinant moduli space on a nodal curve

Let $Y$ be an integral nodal projective curve of arithmetic genus $g\ge 2$ with $m$ nodes defined over an algebraically closed field $k$ and $x$ a nonsingular closed point of $Y$. Let $n$ and $d$ be coprime integers with $n\ge 2$. Fix a line bundle $L$ of degree $d$ on $Y$. Let $U_Y(n,d,L)$ denote the (compactified) "fixed determinant moduli space". We prove that the restriction $\mathcal{U}_{L,x}$ of the Poincare bundle to $x \times U_Y(n,d,L)$ is stable with respect to the polarisation $θ_L$ and its restriction to $x \times U'_Y(n,d,L)$, where $U'_Y(n,d,L)$ is the moduli space of vector bundles of rank $n$ and determinant $L$, is stable with respect to any polarisation. We show that the Poincaré bundle $\mathcal{U}_{L}$ on $Y \times U_Y(n,d,L)$ is stable with respect to the polarisation $a α+ b θ_L$ where $α$ is a fixed ample Cartier divisor on $Y$ and $a, b$ are positive integers.

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arXiv2020

Projective Poincaré and Picard bundles for moduli spaces of vector bundles over nodal curves

Let $U^{'s}_L(n,d)$ be the moduli space of stable vector bundles of rank $n$ with determinant $L$ where $L$ is a fixed line bundle of degree $d$ over a nodal curve $Y$. We prove that the projective Poincare bundle on $Y \times U^{'s}_L(n,d)$ and the projective Picard bundle on $U^{'s}_L(n,d)$ are stable for suitable polarisation. For a nonsingular point $x \in Y$, we show that the restriction of the projective Poincare bundle to $x \times U^{'s}_L(n,d)$ is stable for any polarisation. We prove that for arithmetic genus $g\ge 3$ and for $g=n=2, d$ odd, the Picard group of the moduli space $U'_L(n,d)$ of semistable vector bundles of rank $n$ with determinant $L$ of degree $d$ is isomorphic to $\mathbb{Z}$.

math.AG↗
arXiv2015

Moduli spaces of vector bundles on a singular rational ruled surface

We study moduli spaces $M_X(r,c_1,c_2)$ parametrizing slope semistable vector bundles of rank $r$ and fixed Chern classes $c_1, c_2$ on a ruled surface whose base is a rational nodal curve. We show that under certain conditions, these moduli spaces are irreducible, smooth and rational (when non-empty). We also prove that they are non-empty in some cases. We show that for a rational ruled surface defined over real numbers, the moduli space $M_X(r,c_1,c_2)$ is rational as a variety defined over $\mathbb R$.

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arXiv2012

Brauer group and birational type of moduli spaces of torsionfree sheaves on a nodal curve

Let U^{'s}_{L}(n,d) be the moduli space of stable vector bundles of rank $n$ and fixed determinant L of degree d on a nodal curve Y. The moduli space of semistable vector bundles of rank n and degree $d$ will be denoted by U'_Y(n,d). We calculate the Brauer groups of U^{'s}_{L}(n,d)$. We study the question of rationality of $U^{'s}_{L}(n,d)$ and $U'_Y(n,d)$.

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arXiv2011

Poincare sheaves on the moduli spaces of torsionfree sheaves over an irreducible curve

Let $Y$ be a geometrically irreducible reduced projective curve defined over real numbers. Let $U_Y$ (respectively, $U'_Y$) be the moduli space of geometrically stable torsionfree sheaves (respectively, locally free sheaves) on $Y$ of rank $n$ and degree $d$. Define $χ\, =\, d+n(1-\text{genus}(Y))$, where $\text{genus}(Y)$ is the arithmetic genus. If $2n$ is coprime to $χ$, then there is a Poincare sheaf over $U_Y\times Y$. If $2n$ is not coprime to $χ$, then there is no Poincare sheaf over any nonempty open subset of $U'_Y$.

math.AG↗
arXiv2007

Stable real algebraic vector bundles over a Klein bottle

Let X be a geometrically connected smooth projective curve of genus one, defined over the field of real numbers, such that X does not have any real points. We classify the isomorphism classes of all stable real algebraic vector bundles over X.

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