Abstract. Let ξ be a stable Chern character on P 1 × P 1 , and let M (ξ) be the moduli space of Gieseker semistable sheaves on P 1 × P 1 with Chern character ξ. In this paper, we provide an approach to computing the effective cone of M (ξ). We find Brill-Noether divisors spanning extremal rays of the effective cone using resolutions of the general elements of M (ξ) which are found using the machinery of exceptional bundles. We use this approach to provide many examples of extremal rays in these effective cones. In particular, we completely compute the effective cone of the first fifteen Hilbert schemes of points on P 1 × P 1 .
We study the birational geometry of Hilbert schemes of points on nonminimal surfaces. In particular, we study the weak Lefschetz Principle in the context of birational geometry. We focus on the interaction of the stable base locus decomposition (SBLD) of the cones of effective divisors of X ½n and Y ½n , when there is a birational morphism f : X ! Y between surfaces. In this setting, N 1 ðY ½n Þ embeds in N 1 ðX ½n Þ, and we ask if the restriction of the stable base locus decomposition of N 1 ðX ½n Þ yields the respective decomposition in N 1 ðY ½n Þ i:e:, if the weak Lefschetz Principle holds. Even though the stable base loci in N 1 ðX ½n Þ fails to provide information about how the two decompositions interact, we show that the restriction of the augmented stable base loci of X ½n to Y ½n is equal to the stable base locus decomposition of Y ½n : We also exhibit effective divisors induced by Severi varieties. We compute the classes of such divisors and observe that in the case that X is the projective plane, these divisors yield walls of the SBLD for some cases.
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.
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