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Consider the moduli space $\mathcal{M}_g$ of genus $g$ curves over $\mathbb{C}$. Let $d,r\geq 1$ be integers so that the Brill Noether number $\rho(g,r, d)=g-(r+1)(g-d+r)>0$ . I am mainly interested …

Let $C$ be a smooth curve over complex numbers. Consider the Brill Noether varieties. If $g$ is the genus of $C$. If $r,d$ are positive integers, $$W^r_d=\{A\in Pic^d(C): h^0(A)\geq r+1\},$$ $$G^r_d= …

Let $X$ be a smooth projective surface over $\mathbb{C}$. Let $L$ be an ample line bundle on $X$. Let $|L|_s$ denote the locus of smooth curves in $|L|$. For $C\in |L|_s$ consider the Brill-Noether v …

Let $X$ be a smooth projective variety with Picard number 1 over $\mathbb{C}$. Let $F$ be a coherent sheaf on $X$ such that $c_1(F)$ is algebraically trivial, and hence numerically trivial. Also rank …

Let $X$ be a complex K3 surface and $C$ a smooth curve on $X$ and $A$ a basepoint free line bundle on $C$. Aprodu's paper - Lazarsfeld Mukai bundles and applications says this. We cannot lift the lin …

Let $C$ be a smooth projective curve. Let $A\in Pic(C)$. The Clifford index of $A$ is defined as $$Cliff (A)= deg\,A-2(h^0(A)-1).$$ What does this actually measure. Next the Clifford index of $C$ is …

Let $p:C\rightarrow S$ be a family of curves with $S=\text{Spec}\, R$ where $R$ is a DVR. Suppose that $C$ is smooth and the generic fiber $X_n$ of $p$ is smooth and the special fibre $X_0$ is a reduc …

If $f:X\rightarrow Y$ is a flat family of projective varieties over an algebraically closed field. Suppose that $X$ and $Y$ are irreducible. Let $F$ be a torsion-free coherent sheaf on $X$ of rank $r$ …

I am new to study of abelian varieties. But I need it in my work. Let $X$ be a ppav, say a Jacobian of a genus 2 curve. Let $L$ be a very ample line bundle on $X$. The set $K(L)=\{x\in X : T_x^* L\s …

Let $X$ be an abelian surface over complex numbers. Let $L$ be a totally symmetric line bundle of type $(r,r)$ for $r\geq 2$. The involution $i$ on $X$ gives an action on $H=H^0(X,L)$ thus giving a …

How do we compute the projective resolution of a representation of a quiver with relations. For example consider the Beilinson quiver $B_4$ $. with the relations ­$\{\alpha_j^k\alpha_i^{k-1}=\alpha_i^ …