38 problems

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…

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. Let $D$ be a divisor satisfying the paper's condition; assume $D$ is effect…

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 $p_1,\dots,p_r$ be $r$ very general points in $\mathbb P^2$. For an integral curve $C\subset\mathbb P^2$, write $\operatorname{mult}_{p_i}(C)$ for its multiplicity at $p_i$. Eq…

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

Self-intersection conjecture. One has

Conjecture on asymptotically special nef divisors. The blow-up $X^n_r$ does not contain nef divisors which are asymptotically special.

Let $X_r$ be the blow-up of $[?]$?

Let $\mathbb P=\mathbb P_\Sigma$ be a projective toric surface with Cox ring $\mathbb C[x_1,\dots,x_n]$ and lattice ideal $I_{\mathbb P}$. Let $e$ denote the identity of the torus…

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 $C\subset\mathbb{P}^n$ be a rational normal curve, and let $X_n=\operatorname{Bl}_C\mathbb{P}^n$. For an integer $d$, write $\operatorname{Sec}_d(C)$ for the $d$-secant variety…

Let $X_{n,s}$ be the blow-up of projective $n$-space at $s$ points, and let $D$ be a divisor. For $I\subset\{1,\ldots,n+3\}$ and a secant parameter $t$, define … and … Write…

Let $X=X_{n,s}$ be the blow-up of projective $n$-space at $s$ general points, and let $D=dH-mE_1-\cdots-mE_s$ be a homogeneous divisor, with degree $d$ and common multiplicity $m$.…

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. If $r\geq 3$ and $d>m_1+m_2+m_3$, then

Let $\Sigma$ be the Riemann sphere with $n$ parabolic points, fix generic eigenvalues $\left\{\rho_{i,k}\right\}_{i=0,k=1}^{n,3}$, and let ${\mathcal M}$ be the moduli space of log…

Let $X$ be a composite of blow-ups of $\mathbb P^2$ at points $p_1,\ldots,p_n$ in very general position. A $(-1)$-rational curve is a rational curve with self-intersection $-1$. Th…

Let $X$ be a compact complex manifold with complex dimension $n$, and let $Z\subseteq X$ be a closed complex submanifold of complex codimension $r\geq 2$. Suppose that…

Let $X$ be a smooth variety, let $Z\subset X$ be a complete intersection of $l$ divisors $W_1,\ldots,W_l$ of $X$, and let $\widetilde X$ be the blow-up of $X$ along $Z$. Maulik–Pan…

Nefness criterion. The divisor $D$ is nef if and only if these F-nef inequalities hold.

Let $R_i$ be the relevant subvariety of $\mathbb G(r,n)$, and for $1\leq i\leq r$ let $\alpha_j$ be the successive blow-ups along the smooth centers introduced in the construction.…