12 problems

Let $Y$ be an Enriques surface and let $v=(r,\beta,n)\in H^{\ast}(Y,\mathbb{Z})$ be primitive, with $r$ even and $2\nmid\beta$. Let $M^Y(v)$ be the automatically smooth moduli spac…

The refined Vafa–Witten multiple-cover conjecture. The refined Vafa–Witten invariant satisfies

Let $a\in\{\operatorname{cob},\operatorname{ell}\}$, let $S$ be a smooth projective surface with $b_1(S)=0$ and $p_g(S)>0$, and choose $H,c_1,c_2$ so that there are no rank $2$ str…

Let $a\in\{\operatorname{cob},\operatorname{ell}\}$, let $\underline\beta\in\mathbb Z^4$ satisfy $\beta_1\equiv\beta_2\pmod 2$ and $\beta_3\ge\beta_4-3$, and let…

Let $S$ be a smooth projective surface with $b_1(S)=0$, $p_g(S)>0$, $K_S\ne0$, and only Seiberg–Witten basic classes $0$ and $K_S$. Let $H,c_1,c_2$ be chosen so that there are no r…

Let $S$ be a smooth projective surface with $b_1(S)=0$, $p_g(S)>0$, $K_S\ne0$, and with only Seiberg–Witten basic classes $0$ and $K_S$. Let $H,c_1,c_2$ be chosen so that there are…

Extremal-pair conjecture. Every $\xi$ above the Rudakov $\delta$-surface has an extremal pair, and every controlling exceptional bundle of an extremal pair is contained in two extr…

Effective-divisor spanning conjecture. The method laid out in this paper produces a set of effective divisors spanning the effective cone of $M(\xi)$ for every $\xi$ above Rudakov'…

Let $P_q(X)=\sum_{i\geq0}\dim H_i(X)q^{i/2}$ be the Poincare polynomial, and consider the rank-two moduli spaces ${\mathcal M}(2,n)$ with the two fixed components corresponding to…

Let ${\overline w}\in{\mathbb Z}^r$, let $\alpha,\beta\geq 1$ satisfy $0\leq w_i<\alpha+\beta$ and $\gcd(\alpha,\beta)=1$, and let $a_i=\sharp\{j\mid w_j=i\}$. Let $\alpha'$ be the…

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…