70 problems

Cao–Jiang conjecture. For any exceptional object $\mathcal{E}\in D(X)$, there exists an autoequivalence $\Phi\in\operatorname{Auteq}(D(X))$ such that $\Phi(\mathcal{E})$ is an exce…

Strong rational connectedness conjecture. The smooth locus $X^{sm}$ of a log Del Pezzo surface $(X,\Delta)$ is strongly rationally connected.

Let $S\subseteq\mathbb{P}^d$ be a possibly singular del Pezzo surface, let $U\subseteq S$ be obtained by deleting exceptional divisors, and let $\rho_S$ be the relevant Picard rank…

Generalized Eckardt point conjecture. If the total log canonical threshold of $X$ is $\frac{n-1}{n}$, then a wild tiger $S$ for $X$ is a cone in $\mathbb{P}^{n-1}$ over a smooth hy…

Let $X$ be either the Hirzebruch surface $\mathbb F_2$ or the surface $B'$ obtained by blowing up six points on a conic, and let $E$ be its $(-2)$-curve. Let $D$ be an effective di…

Let $S$ be a quasi-smooth, well-formed del Pezzo hypersurface of degree $d$ in the weighted projective space … where $S$ is defined over an algebraically closed field $\Bbbk$ of ch…

Let $M$ be a del Pezzo surface of degree $1$, $5$, or $7$, and let $D$ be a generic smooth representative of $M$'s anticanonical divisor. A stable $\mathbb{R}$-polarization of $X$…

Let $S$ be a log del Pezzo surface. The cylindrical ample cone conjecture. If the cylindrical ample cone … contains the anticanonical class, then … Equivalently, the cylindrical am…

Quartic near-miss infinitude conjecture. If this equation has an integer solution, then it has infinitely many integer solutions. The conjecture is suggested by the paper's non-thi…

Let $A,B\in\mathbb{Z}$ satisfy $4A^3-27B^2\neq 0$, let $Q(u,v)$ be a positive-definite binary quadratic form, and define … For rational points represented by integral $(x,y,u,v)$ w…

Frobenius uniqueness conjecture. A Wahl chain $\texttt{W}$ in $\mathbb{W}_9$ is determined by its index.

Let $S$ be a del Pezzo surface, let $E$ be a suitable anticanonical divisor, and let $M_{\beta}$ be the moduli space of stable one-dimensional sheaves of class $\beta$ and Euler ch…

Let $\overline{X}$ be a singular del Pezzo surface. The height bound conjecture. If $\operatorname{char}\mathscr{k}\neq 2,3$, then … The height is a bounded invariant intended to o…

Let $S$ be a log del Pezzo surface, meaning a del Pezzo surface with at worst klt singularities. Let … be its cylindrical ample set, where $\mathrm{Amp}(S)$ is the ample cone. Chel…

Let $S$ be a smooth, affine, cubic surface whose automorphism group is infinite and discrete. A classification conjecture asserts that $S$ is either a Markov-type surface, or the c…

Refined BPS stabilization conjecture. The refined BPS number $n_\beta^{i,j}$ stabilizes when $\beta$ is sufficiently positive. This conjectures asymptotic stability of the refined…

Let $S$ be a del Pezzo surface, let $H$ be the polarization, let $\beta$ be an effective divisor, let $\chi\in\mathbb{Z}$, and let $M_{\beta,\chi}$ be the moduli space of one-dimen…

Let $M_{d,\chi}$ be the moduli space associated with the curve class $\beta=dH$ and Euler characteristic $\chi\in\mathbb{Z}$ on $\mathbb{P}^2$, where $H$ is the class of a line and…

Let $M$ be a rigid del Pezzo surface. An index zero $(p,q)$-unicuspidal rational symplectic curve in $M$ is as in the source, and an almost toric fibration is a fibration…

Let $Y$ be a del Pezzo surface, let $(\mathcal X,W)$ be its Landau–Ginzburg mirror, let $\alpha:K(Y)\to H_2(\mathcal X,W^{-1}(+\infty);\mathbb Z)$ be the map from the preceding cen…

Let ${\mathcal F}^\ast$ be the locally symmetric variety, let $\lambda$ be its Hodge line bundle, and for $s\in[0,1]\cap{\mathbb Q}$ define … Let $P^\ast$ denote the relevant open…

Transverse-distance conjecture. The transverse distance between the D3-brane probe and the $E_n$ 7-branes is proportional to $\operatorname{vol}(V_2)$ in the local Calabi–Yau three…

Let $S$ be a Gorenstein surface with ample anticanonical sheaf, let $D$ be a polarization of $S$, and let $n^D_\gamma$ be the local BPS invariant associated with a curve class…

Let $\mathbb{B}_k$, for $0 \leq k \leq 8$, and $\mathbb{C}\mathbb{P}^1 \times \mathbb{C}\mathbb{P}^1$ be the indicated series of del Pezzo surfaces. Let $\mathcal{L}^{k}_c S^4$, fo…

Let a real symplectic del Pezzo surface of degree $1$ be equipped with a $J$-holomorphic structure satisfying the Bertini condition. Algebraicity conjecture. Each deformation class…