279 problems

Let $S$ be a smooth surface, and let $C$ be a reduced and irreducible curve on $S$. Bounded Negativity Conjecture. There is a constant depending only on $S$ such that … is bounded…

Let $X$ be a surface, let $\mathcal{L}$ be a line bundle on $X$, and let $\delta$ be a non-negative integer. Let $N_\delta^X(\mathcal{L})$ be the number of curves in…

Let $S$ be a smooth projective surface defined over $\bar{\mathbb Q}$. Denote by $\operatorname{CH}_0(S)^0$ the Chow group of zero-cycles of degree zero, by $\operatorname{Alb}(S)$…

Nagata–Biran–Szemberg conjecture.

Let $X$ be an algebraic surface of general type, and let its index be positive, meaning that $c_1^2(X)-2c_2(X)>0$. Bogomolov's conjecture. The fundamental group $\pi_1(X)$ should b…

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 $(X_n)_{n\geq 1}$ be the tower of normal projective surfaces obtained by successive $\mathbb{Z}/3$-Galois covers, and let $\widetilde{X}_n\to X_n$ be the smooth minimal resolut…

Let $X$ be a projective and simply connected M-surface, and write $\mathbb{R}X$ and $\mathbb{C}X$ for its real and complex loci. Let $\beta_1(\mathbb{R}X)$ denote the first Betti n…

Let $f:S\to B$ be a fibration of a smooth projective surface over a smooth projective curve, with general fibre of genus $g$, and let $q_f:=q(S)-b$ denote its relative irregularity…

Let $D=D_1+\dots+D_n$ be a reduced curve with simple normal crossings on a smooth algebraic surface $V$, and let $\Gamma$ be the dual graph of $D$. For each component, set…

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 … 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…

Weak Bounded Negativity Conjecture. There is a constant $b_{X,g(C)}$, depending only on $X$ and $g(C)$, such that

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…

Harris's conjecture. The locus $\operatorname{NL}_d$ has only finitely many exceptional components.

Let $R$ be the graded section ring in the preceding construction, and let $v_3,v_4,v_5,v_6$ denote the generators of the module in the indicated weights $(1,4)$ and $(1,5)$. Degree…

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 fake projective plane, and let $L$ be a cubic root of its canonical bundle, so that $L^{\otimes 3}\cong K_X$, assuming such a root exists. Acyclicity conjecture. The l…

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…

Phantom-category conjecture. The derived categories of the Barlow surface and of the rational blow-down described in the source contain a phantom category as a semiorthogonal piece…

Griffiths–Harris conjecture. For $3 \le r \le d$, every irreducible component of $\operatorname{NL}_{r,d}$ has codimension at least