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Let $��_1:\mathcal{X} \to ��$ be a flat family of smooth, projective curves of genus $g \ge 2$, degenerating to an irreducible nodal curve $X_0$ with exactly one node. Fix an invertible sheaf $\mathcal{L}$ on $\mathcal{X}$ of relative odd degree. Let $��_2:\mathcal{G}(2,\mathcal{L}) \to ��$ be the relative Gieseker moduli space of rank $2$ semi-stable vector bundles with determinant $\mathcal{L}$ over $\mathcal{X}$. Since $��_2$ is smooth over $��^*$, there exists a canonical family $\widetilde��_i:\mathbf{J}^i_{\mathcal{G}(2, \mathcal{L})_{��^*}} \to ��^{*}$ of $i$-th intermediate Jacobians i.e., for all $t \in ��^*$, $(\widetilde��_i)^{-1}(t)$ is the $i$-th intermediate Jacobian of $��_2^{-1}(t)$. There exist different N��ron models $\overline��_i:\overline{\mathbf{J}}_{\mathcal{G}(2, \mathcal{L})}^i \to ��$ extending $\widetilde��_i$ to the entire disc $��$, constructed by Clemens, Saito, Schnell, Zucker and Green-Griffiths-Kerr. In this article, we prove that in our setup, the N��ron model $\overline��_i$ is canonical in the sense that the different N��ron models coincide and is an analytic fiber space which graphs admissible normal functions. We also show that for $1 \le i \le \max\{2,g-1\}$, the central fiber of $\overline��_i$ is a fibration over product of copies of $J^k(\mathrm{Jac}(\widetilde{X}_0))$ for certain values of $k$, where $\widetilde{X}_0$ is the normalization of $X_0$. In particular, for $g \ge 5$ and $i=2, 3, 4$, the central fiber of $\overline��_i$ is a semi-abelian variety. Furthermore, we prove that the $i$-th generalized intermediate Jacobian of the (singular) central fibre of $��_2$ is a fibration over the central fibre of the N��ron model $\overline{\mathbf{J}}^i_{\mathcal{G}(2, \mathcal{L})}$. In fact, for $i=2$ the fibration is an isomorphism., to appear in Revista Matem\'atica Complutense

Fix $n\geq 5$ general points $p_1, \dots, p_n\in\mathbb{P}^1$, and a weight vector $\mathcal{A} = (a_{1}, \dots, a_{n})$ of real numbers $0 \leq a_{i} \leq 1$. Consider the moduli space $\mathcal{M}_{\mathcal{A}}$ parametrizing rank two parabolic vector bundles with trivial determinant on $\big(\mathbb{P}^1, p_1,\dots , p_n\big)$ which are semistable with respect to $\mathcal{A}$. Under some conditions on the weights, we determine and give a modular interpretation for the automorphism group of the moduli space $\mathcal{M}_{\mathcal{A}}$. It is isomorphic to $\left(\frac{\mathbb{Z}}{2\mathbb{Z}}\right)^{k}$ for some $k\in \{0,\dots, n-1\}$, and is generated by admissible elementary transformations of parabolic vector bundles. The largest of these automorphism groups, with $k=n-1$, occurs for the central weight $\mathcal{A}_{F}= \left(\frac{1}{2},\dots,\frac{1}{2}\right)$. The corresponding moduli space ${\mathcal M}_{\mathcal{A}_F}$ is a Fano variety of dimension $n-3$, which is smooth if $n$ is odd, and has isolated singularities if $n$ is even., 13 pages

Mumford and Newstead generalized the classical Torelli theorem to higher rank i.e., a smooth, projective curve $X$ is uniquely determined by the second intermediate Jacobian of the moduli space of stable rank $2$ bundles on $X$, with fixed odd degree determinant. In this article we prove the analogous result in the case $X$ is an irreducible nodal curve with one node. As a byproduct, we obtain the degeneration of the second intermediate Jacobians and the associated N\'{e}ron model of a family of such moduli spaces., Comment: to appear in Documenta Mathematica vol. 24, p. 1739-1767

Let $R$ be a complete discrete valuation ring with fraction field of characteristic $0$ and algebraically closed residue field of characteristic $p>0$. Let $X_R \to \mathrm{Spec}(R)$ be a smooth projective morphism of relative dimension $1$. We prove that, given a line bundle $\mathcal{L}_R$ the moduli space of Gieseker stable torsion-free sheaves of rank $r\geq 2$ over $X_R$, with determinant $\mathcal{L}_R$, is smooth over $R$., 14 pages, To appear in Proceedings of the conference on "Analytic and Algebraic Geometry of Bundles"

In this article we study certain specialization properties of the index of varieties defined by Koll\'{a}r., Comment: Some of the results needed additional hypothesis. We prove a slightly different, but more general result in the article titled "Examples of varieties with index one on $C_1$-fields"