123 problems

Let $U$ be a smooth algebraic surface, let $p_g(U)$ denote its geometric genus, and let $T(U)$ be the Albanese kernel, namely the kernel of the Albanese morphism…

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 $S$ 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…

Chisini's conjecture. Given two generic coverings $f:S\longrightarrow\mathbb{P}^2_{\mathbb{C}}$ and $f':S'\longrightarrow\mathbb{P}^2_{\mathbb{C}}$, both of degree at least $5$, wi…

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 $X$ be a smooth projective surface with a fixed Kähler class $[\omega_0]$ and a line bundle $L$. The stability condition $\sigma_{[\omega_0]}$ is the stability condition suppor…

Let $S$ and $T$ be smooth projective surfaces. Let $\Gamma\in\mathrm{CH}^2(S\times T)$ be a codimension-$2$ cycle, and let … be the induced map. Define…

Let $S$ be a rational homology projective plane, meaning a normal projective complex surface whose Betti numbers are the same as those of $\mathbb{CP}^2$. Assume that $S$ has quoti…

Let $S$ be a smooth projective surface. A symplectic automorphism is an automorphism acting trivially on the space of holomorphic two-forms $H^{2,0}(S)$. It acts on the Albanese ke…

Let $X$ be the blow-up of $\mathbb P^2$ at $n$ generic points, and let $C$ be a curve on $X$. The SHGH vanishing conjecture asserts that … for every curve $C$ on $X$. The source id…

Weighted Bounded Negativity Conjecture. There exists a nonnegative integer $b_w\in\mathbb Z$ such that

Let $X$ be a smooth projective surface and let $L$ be a nef divisor on $X$. Let $\epsilon(n;X,L)$ denote the relevant multipoint Seshadri constant for $n$ points. Seshadri constant…

Let $X$ be a minimal surface of general type and maximal Albanese dimension. The Severi equality characterization states that … if and only if the canonical model of $X$ is a flat…

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 $S$ 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…