Nef Cones of Nested Hilbert Schemes of Points on Surfaces

Abstract

Let X be the projective plane, a Hirzebruch surface, or a general K3 surface. In this paper, we study the birational geometry of various nested Hilbert schemes of points parameterizing pairs of zero-dimensional subschemes on X. We calculate the nef cone for two types of nested Hilbert schemes. As an application, we recover a theorem of Butler on syzygies on Hirzebruch surfaces. 1 NEF CONES OF NESTED HILBERT SCHEMES OF POINTS ON SURFACES 2 product of projective lines, on Hirzebruch surfaces, and on del Pezzo surfaces of degree at least two [BC13]. Recently, there has been tremendous progress using Bridgeland stability to compute the nef cones of Hilbert schemes of points on K3 surfaces [BM14b], Abelian surfaces ([MM13], [YY14]), Enriques surfaces [Nue16], and all surfaces with Picard number one and irregularity zero [BHL + 16].To calculate the nef cone for nested Hilbert schemes, we must first understand the Picard group and the Néron-Severi group.Proposition A. Let X be a smooth projective surface of irregularity zero and fix n ≥ 2. ThenIn particular, the Néron-Severi group N 1 (X [n+1,n] ) has rank 2(ρ(X) + 1), where ρ(X) is the picard number of X.Knowing the Picard groups, we can describe the nef cones. To easily state our theorem, let us first recall the nef cone of the Hilbert schemes of points on P 2 . The nef cone of P 2[n] is spanned by the two divisorswhere H[n] is the class of the pull-back of the ample generator via the Hilbert-Chow morphism and B[n] is the exceptional locus.Theorem B. The nef cone of P 2[n+1,n] , n > 1, is spanned by the four divisors, and pr * a (D n [n + 1]). Similar results hold for the Hirzebruch surfaces F i , i ≥ 0, and general K3 surfaces S as well as for the universal families on all of these surfaces.Knowing the nef cone allows us to recover a theorem of Butler about projective normality of line bundles on Hirzebruch surfaces [But94].Proposition C. Let X be a Hirzebruch surface, and A be an ample line bundles on X, then L = K X + nA is projectively normal for n ≥ 4.

The Hilbert scheme of a pair of linear spaces is a Mori dream space

“…Chen, Coskun and Nollet [CCN11] showed that the component corresponding to a pair of codimension two linear spaces meeting transversely in P n is a Mori dream space. For more examples refer to [GHS18] and [RY20].…”