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Suppose $C$ is a genus $4$ smooth projective curve over complex numbers. The Abel-Jacobi map from $Sym^4(C)$ to $Jac(C)$ is birational. Is this a blow-up along a surface or a curve. Can one determine …

Consider the tautological bundle $S$ on a Grassmannian $G(r,n)$ of $r$-subspaces in $\mathbb{C}^n$. Is $S$ trivial outside (large degree) hypersurfaces on $G(r,n)$. Morel's theorem seems to confirm wh …

I was looking for properties of reflexive sheaves on a variety. Suppose $X\rightarrow Y$ is etale outside codimension two subset. Say $X$ is smooth. Is the pullback of a reflexive sheaf on $Y$, refle …

suppose $p:X\times Y \rightarrow Y$ is a second projection. Let $F$ be a coherent sheaf on $X\times Y$. Then $H^0(Y,R^1p_*F)$ is part (i.e. fits in a long exact sequence) of $H^1(X\times Y, F)$, using …

Suppose $X$ is a smooth variety and consider a finite group $G$ acting on $X$. assume that the quotient map $X\rightarrow X/G$ is etale outside a codimension two subset. Suppose $H$ is coherent sheaf …

Suppose $X$ is a smooth variety and $G$ is a finite group acting on $X$. $X/G$ is not locally factorial. Let $h: X\rightarrow X/G$ be the quotient morphism. Suppose there is a coherent sheaf $F'$ on …

Suppose $C$ is a smooth projective curve over complex numbers. The singularities of the theta divisor $\Theta$ in $Pic^{g-1}(C)$ is described in the literature. It is $W^{1}_{g-1}=\{l\in Pic^{g-1}(C): …

Suppose $X$ is a smooth projective variety over a field $k\subset \mathbb{C}$. Let $CH^r(X)_{hom}$ be the Chow group of codimension $r$ cycles defined over $k$ and homologous to zero. The usual Abel-J …

Suppose $G$ is a Grassmannian variety. Let $T_G$ be the tangent bundle. Then $H^0(G,T_G)$ is non zero. I wondered if one knows $H^0(G,\wedge^iT_G)$, for $i>0$. THanks

Suppose $X$ is any quasi-projective variety. let $K^0(X)$ denote the Grothendieck group of locally free sheaves. Suppose $U$ is an open subset of $X$. Is there a localization sequence: $$ K^0(X)\ri …

I am trying to understand the structure of Moduli space of rank two parabolic bundles on $\mathbb{P}^1$, of degree zero or degree one, with weights $(\frac{1}{2},\frac{1}{2},\dotsc\frac{1}{2})$ at $n$ …

Given a coherent sheaf $F$ on a smooth variety $X$, we know that $F$ is locally free on an open subset $U$ in $X$ outside a codimension two subset. Say the rank is $k$. Is there a locally free sheaf …