17 problems

Strange duality conjecture. For any $g\geq6$, this map is an isomorphism.

Let $k\geq 1$, and consider the Hilbert scheme $(\mathbb P^2)^{[k]}$ together with the appropriate theta bundle arising from the strange-duality construction. Let the expected size…

Strange Duality conjecture. The map $D$ is an isomorphism. This conjecture predicts a duality between spaces of sections of determinant line bundles on moduli spaces of semistable…

Let $X$ be a smooth projective curve of genus $g$, let $SU_X(r)$ denote the moduli space of semistable rank-$r$ vector bundles with trivial determinant, and let $U_X(k,0)$ denote t…

Verlinde–Segre correspondence. The universal series satisfy

Let $\mathbb P(\mathcal G)$ be the parameter space carrying the universal family $\mathcal E$, and let $I_2$ be the ideal sheaf of a length-two subscheme of $\mathbb P^2$. The thet…

Assume a general sheaf in $M(e)$ is locally free, let $V$ have Chern character $e^*+f$, and assume $\chi(e\cdot f)=0$. Let $\operatorname{Quot}(V,f)$ parametrize quotients…

Rank-three Verlinde–Segre prediction. Setting

Universal-series correspondence. The series satisfy

Theta-bundle pushforward conjecture. If $L$ is sufficiently ample, then $\pi^{L,c_1}_*(\lambda(\Theta_{2,c_1}))\otimes\mathcal O_{|L|}=p^{X,c_1}_L(\mathcal O_{|L|}(-1))$ and

Let $c,v\in \mathrm{K}(\mathbb{P}^{2})$ satisfy $\chi(c\cdot v)=0$, and let $\mathrm{M}(c)$ and $\mathrm{M}(v)$ be the corresponding moduli spaces of semistable sheaves. For a line…

Let $c,v\in \mathrm{K}(\mathbb{P}^{2})$ satisfy $\chi(c\cdot v)=0$. Let $\mathrm{M}(c)$ and $\mathrm{M}(v)$ be the moduli spaces of semistable sheaves with topological types $c$ an…

Let $X$ be a smooth complex projective surface with $H$ an ample divisor. Let $u=(0,L,\chi(u)=0)$, where $L$ is an effective line bundle on $X$, and let…

Let $X$ be a simply connected elliptic surface with a section and at worst nodal fibers, and let $v$ and $w$ be orthogonal topological types of ranks $r,s\geq 2$ satisfying … and ……

Beauville's symplectic strange-duality conjecture. The morphism $\mathsf {D}$ is an isomorphism. Beauville established equality of the dimensions of the two spaces of sections, but…

Let $v$ be a Mukai vector, and let $F_{1}$ and $F_{2}$ be sheaves with the same Mukai vector orthogonal to $v$. Let $\Theta_{F_i}$ denote the corresponding theta line bundles on th…

Let $A$ be the abelian surface and let $v$ and $w$ be Mukai vectors satisfying Assumptions and, with the associated moduli spaces and determinant line bundles as in the paper. Deno…