75 problems

Let $q_1,\ldots,q_n$ be independent generic points in $\mathbb{P}_{\mathbb{C}}^2$, with $n\geq 10$. For a plane curve of degree $d$, suppose that it passes through each $q_i$ with…

Let $S$ be the blow-up of $\mathbb{P}^2$ at $k \geq 1$ general points, and let $L$ be an effective and nef line bundle on $S$. Harbourne–Hirschowitz conjecture. One has … This vani…

Nagata–Biran–Szemberg conjecture.

Self-intersection conjecture. One has

Let $\mathbb F_{e,r}$ be the blow-up of the Hirzebruch surface $\mathbb F_e$ at $r$ very general points. Let $D$ be a divisor satisfying the paper's condition; assume $D$ is effect…

Let $\mathbb{F}_{e,r}$ denote the blow-up of the Hirzebruch surface $\mathbb{F}_e$ at $r$ very general points $p_1,\dots,p_r$. A curve is irreducible and reduced if it is an irredu…

Let $X=X_{2,s}$ be the blow-up of the projective plane at $s$ general points, and let $D\text{ in }\operatorname{Pic}(X)$ be an effective divisor of the specified form. Write…

SHGH conjecture. Either

Let $Y\to X$ be a symplectic embedding of a nonformal simply connected manifold $Y$ into a simply connected compact manifold $X$. Blow-up conjecture. The blow-up of $X$ along $Y$ i…

All-genus blow-up invariance conjecture. The genus condition $g\leq 1$ in the stated equality is technical: for every genus $g$,

Cone-of-curves formulation. The extremal rays of $\mathcal{B}_1$ that do not lie on $\operatorname{Nul}(Y)$ are spanned by classes of $(-1)$-curves, and

Negative-curve conjecture. The curve $C$ is a $(-1)$-curve of $Y$, meaning that it is a smooth rational curve with self-intersection $-1$.

Let $\phi\colon\widehat V\to V$ be a blowing down between nonsingular complex projective varieties, with $\widehat V$ obtained by blowing up $V$ along a nonsingular irreducible cen…

Self-intersection–genus conjecture. One has

Let $Z\subset V$ be a nonsingular complete intersection, let $\widetilde{V}$ be the blow-up of $V$ along $Z$, and let $H^*(V,\mathbb{Q})\rightarrow H^*(Z,\mathbb{Q})$ be the classi…

Let $X_r^3$ be the blow-up of $\mathbb{P}^3$ at $r$ points in general position. For a special effective divisor $D$ on $X_r^3$, let $B$ be an irreducible and reduced positive-dimen…

Let $X_r^n$ be the blow-up of $\mathbb{P}^n$ at $r$ points in general position, let $D$ be a special effective divisor on $X_r^n$, and let $B$ be an irreducible and reduced positiv…

Ciliberto's conjecture. (i) The only negative curves on $X$ are $(-1)$-curves. (ii) The system $\mathbb{L}$ is special if and only if there is a $(-1)$-curve $C$ on $X$ correspondi…

Negative curve conjecture. Every negative curve on $X_n$ is a $(-1)$-rational curve.

Let $\mathbb F_{e,r}$ be the blow-up of the Hirzebruch surface $\mathbb F_e$ at $r$ very general points, and let $\widetilde{C}_e$ be the strict transform of the negative section…

Let $\mathbb F_{e,r}$ be the blow-up of the Hirzebruch surface $\mathbb F_e$ at $r$ very general points, and let $C_e$ be the negative section of $\mathbb F_e$ with strict transfor…

Let $X={\mathsf{Bl}}_{\mathbb{P}^s}(\mathbb{P}^r)$, let $C$ be a general curve of genus $g$, and let $\widetilde{Q}_\beta(C,X)$ be the moduli space of complete quasimaps $u:C\to X$…

Let $Y$ be a Fano manifold, and let $Z$ be an irreducible smooth Fano subvariety of $Y$ with … Let $X=\operatorname{Bl}_Z Y$ be the blow-up of $Y$ along $Z$, and denote by…

Let $X$ satisfy the hypotheses of the Bogomolov–Miyaoka–Yau conjecture for four-manifolds with a Seiberg–Witten basic class, and let … be its smooth blow-up. Let $(E,H)$ be a compl…

Stronger bounded negativity conjecture. If $C$ is an integral curve in $\operatorname{Bl}_{n}\mathbb{P}^2$, then