Effective divisors and Newton-Okounkov bodies of Hilbert schemes of points on toric surfaces

Abstract

We compute the (unbounded) Newton-Okounkov body of the Hilbert scheme of points on ${\mathbb{C}}^2$. We obtain an upper bound for the Newton-Okounkov body of the Hilbert scheme of points on any smooth toric surface. We conjecture that this upper bound coincides with the exact Newton-Okounkov body for the Hilbert schemes of points on ${\mathbb{P}}^2$, ${\mathbb{P}}^1\times {\mathbb{P}}^1$, and Hirzebruch surfaces. These results imply upper bounds for the effective cones of these Hilbert schemes, which are also conjecturally sharp in the above cases.

41 pages, 3 figures