Extremal divisors in the Hilbert scheme of points on $\mathbb{P}^{2}$ are preserved under residuality
Montserrat Vite
TL;DR
This work addresses whether extremal divisors in the Hilbert scheme of $n$ points on $\mathbb{P}^{2}$ are preserved under residuality, focusing on triangular numbers $n=\tfrac{r(r+1)}{2}$ and tangential numbers $n=2r(r+1)$. Building on the known birational geometry of $\mathbb{P}^{2[n]}$ and the effective cone generated by the divisor classes $B$ and $D_r$, the authors define a residual operator $\mathscr{L}$ via complete intersections of curves and prove $\mathscr{L}D_{r-1}=D_r$ in the triangular case and $\mathscr{L}D_r=D_{r+1}$ in the tangential case, using cohomological equalities and incidence-geometry counts. A key result is that a point $Z$ lies in an extremal divisor if and only if its residual $Z'$ lies in the corresponding extremal divisor, yielding a precise preservation of extremality under residuality. This sheds light on a symmetry in the birational landscape of $\mathbb{P}^{2[n]}$ and provides a mechanism to transport extremal divisors across Hilbert schemes with related $n$, informing stable base locus decompositions and potential birational models.
Abstract
Let $n=\frac{r(r+1)}{2}$ or $n=r(r+1)$. We prove that the property of being extremal is preserved under residuality on the Hilbert scheme of $n$ points in the plane.