7 problems

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…

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…

Harbourne-Hirschowitz conjecture. The system $\mathscr{L}(d,n,m)$ is non-special, meaning that it has expected dimension $\max\{-1,v\}$. This predicts a precise dimension for homog…

Harbourne-Gimigliano-Hirschowitz conjecture. The linear system $|D|$ has expected dimension

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 $X$ be the blow-up of ${\mathbb P}^n$ at $n+3$ general points, and let $D$ be a divisor on $X$ as in the source. Let $\operatorname{\sigma ldim}(D)$ denote the secant linear vi…