123 problems

Let $X$ be a smooth complex projective surface, and let $C$ be a reduced curve on $X$. Bounded Negativity Conjecture. For each such $X$, there exists a number $B_X\geq0$, depending…

Let $E$subset W be the special fiber consisting of a nodal curve on an algebraic surface, let $\mathop{\operatorname{Def}}_{(E \subset W)}$ be the versal deformation space of the p…

Let $X$ be a smooth projective surface, and let $C$ be a curve on $X$. Write $h^i(\mathcal O_X(C))=\dim H^i(X,\mathcal O_X(C))$. The bounded cohomology conjecture asserts that ther…

Virtual Vafa–Witten formula. $\overline{\chi}_{-y}^{\mathrm{vir}}(M)$ equals the coefficient of $x^{\operatorname{vd}(M)}$ in

The nef anticanonical bundle converse. If the anticanonical bundle $-K_X$ of $X$ is nef, then for some finite extension $K'$ of $K$ the set $X(K')$ of $K'$-rational points of $X$ i…

Integral-Tango characterization. If there is a counterexample to the Kodaira vanishing theorem on $Y$, then there exists a fibration

Tango-curve characterization. If there is a counterexample to the Kawamata–Viehweg vanishing theorem on $X$, then there exists a dominant rational map

Let $f\colon S \longrightarrow B$ be a fibration of genus $g$, and let $q_f$ denote its relative irregularity, with $q_f<g-1$. The slope $s(f)$ is the ratio of the relevant relativ…

Artin–Shioda conjecture. Every supersingular $K3$ surface is unirational.

Let $S$ be a rational homology $\mathbb C\mathbb P^2$ with quotient singularities, and write … Assume that $S^0$ is simply connected. Simply connected smooth-locus conjecture. Then…

Let $X$ be a simply connected algebraic surface, let $D\subset\mathbb{CP}^2$ be the branch curve of the relevant projection, and let $G^0$, $K$, $\bar{G}$, $\phi$, $\Lambda$, and…

Segre's conjecture. If $\mathcal L$ is non-empty and reduced, then it is non-special.

Let … be a 5-dimensional Seifert bundle with $L$ smooth. Assume that $L$ is a simply connected rational homology sphere. Rationality conjecture. Then $S$ is a rational surface. Thi…

Let $S$ be a projective surface with quotient singularities such that … and let $S^0$ denote its smooth locus. Assume that $S^0$ is simply connected. Rationality conjecture. Then…

The resultant formula. If $V_1(A)=\emptyset$ and $V_2(A)\ne\emptyset$, then, up to a nonzero constant,

Let $X$ be a simply connected complex surface, let $f_k:X\to\mathbb{CP}^2$ be a generic projection defined by sections of $L^{\otimes k}$, let $\tilde{X}_k$ be the associated Galoi…

Local cohomology conjecture. Under the same notation and assumptions,

Let $S$ be a very general K3 surface. A dominant rational map from the product of two smooth projective curves to $S$ is a rational map $C\times D\dashrightarrow S$ whose image is…

Let $S$ be a rational homology projective plane with quotient singularities, and write … The smooth locus $S^0$ is assumed to be simply connected. Kollár's algebraic Montgomery–Yan…

Let $X$ be a smooth complex projective surface, let $B,\omega\bigin\operatorname{NS}(X)\otimes\mathbb{R}$ with $\omega$ ample, and let…

Let $X$ be a Coble surface with Coble curve $C\simeq\mathbb{P}^1$, and let … be the restriction homomorphism. Coble's restriction-map problem. Is $\ker(\rho)$ trivial for a general…

Let $n$ be a nonnegative integer, set $d(n)=6+\max\\{3,n\\}$, and let $X$ be obtained by blowing up $\mathbf{F}_n$ at $d(n)$ general points. Let $\mathcal{C}_n$ be a subcategory of…

Let $\pi\colon X\rightarrow\operatorname{Spec} k$ be a smooth complete surface over a perfect field $k$ of characteristic $p\neq0$. Let $H_{\mathrm{fl}}^i(X,\mu_p)$ denote flat coh…

Let $X$ be a fake projective plane over an algebraically closed field of characteristic $p>0$, meaning a smooth projective surface with the same Betti numbers as the projective pla…

Let $f:X\dashrightarrow C$ be a canonically fibered surface, meaning that $X$ and $C$ are irreducible, smooth and projective over $\mathbb{C}$ with dimensions $2$ and $1$, respecti…