133 problems

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…

Laface–Ugaglia conjecture. The special effect varieties for Cremona reduced linear systems in $\mathbb{P}^3$ are lines through two points and quadrics through nine points.

Let $X$ be a smooth projective variety of dimension $n\geq 2$, let $K_X$ be its canonical line bundle, and let $L$ be an ample line bundle on $X$. Moraga–Yeong's conjecture. The co…

Donagi–Morrison conjecture. There exists a line bundle $M\in\operatorname{Pic}(S)$ adapted to $|H|$ such that $|A|$ is contained in the restriction of $|M|$ to $C$ and

Hirschowitz–Harbourne conjecture. The system $L$ is special if and only if there exists a $-1$-curve $C\subset\mathbb{P}^{2}$ such that

Let $S$ be a rational surface, let $D$ be an ample, non-special divisor on $S$, and let $p_1,\ldots,p_r$ be $r$ points on $S$. Write … for the corresponding linear system, and call…

Cohomological special effect conjecture. The system ${\mathcal{L}}$ is special if and only if it is cohomologically special.

Numerical special effect conjecture. The system ${\mathcal{L}}_{2,d}(-\sum_{i=1}^hm_iP_i)$ is special if and only if it is numerically special.

Hirschowitz's conjecture. The only special linear systems are the $(-1)$-special ones.

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…

Weyl expected dimension conjecture. For every effective divisor $D$ on $\widehat{X^4_8}$, the dimension of its space of global sections equals its Weyl expected dimension:

Laface–Ugaglia conjecture. The dimension of the space of global sections of an effective divisor on $X^3_s$ depends on the multiplicities of containment in its base locus of Weyl l…

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…

Let $({\mathcal C},\boldsymbol{r})$ be a consistent finite weighted configuration of infinitely near points of $\mathbb{P}^2$, where $r_p$ is the multiplicity assigned to $p$, and…

SHGH conjecture. Either

Let $Y$ be a generic $K3$ surface, and let $\mathcal{L}$ be a nonempty and reduced linear system on $Y$. Segre's conjecture for generic K3 surfaces. The linear system $\mathcal{L}$…

A linear system $\mathcal{L}$ is (-1) special if there are pairwise disjoint (-1) curves $A_1,\dots,A_r$ such that $\mathcal{L}\cdot A_j=-N_j$ with $N_j\geq 1$ for every $j$ and…

Let $D\subset\mathbb{N}^{2}$ be finite, and consider a system $\mathcal{L}_{D}(m_{1},\dots,m_{r})$. The system admits a reduction algorithm if there is a sequence of sets … that sa…

Let $X$ be a K3 surface with $H^2=n\in2{\mathbb{Z}}$, and let ${\mathcal{L}}={\mathcal{L}}^n(d,m_1,\dots,m_h)$ be the system of curves in $|dH|$ through general points…

Let ${\mathcal{L}}=|ah+bF-\sum_{i=1}^{h}m_iP_i|$ be a system on the Hirzebruch surface ${\mathbb{F}}_e$, and let ${\mathcal{M}}$ be its residual system after repeatedly subtracting…

Laface's conjecture. With $\mathcal L$ and $S$ as above:

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

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…