Abstract. The space of smooth rational cubic curves in projective space P r (r ≥ 3) is a smooth quasi-projective variety, which gives us an open subset of the corresponding Hilbert scheme, the moduli space of stable maps, or the moduli space of stable sheaves. By taking its closure, we obtain three compactifications H, M, and S respectively. In this paper, we compare these compactifications. First, we prove that H is the blow-up of S along a smooth subvariety which is the locus of stable sheaves which are planar (i.e. support is contained in a plane). Next we prove that S is obtained from M by three blow-ups followed by three blow-downs and the centers are described explicitly. Using this, we calculate the cohomology of S.
The space of smooth rational curves of degree d in a projective variety X has compactifications by taking closures in the Hilbert scheme, the moduli space of stable sheaves or the moduli space of stable maps respectively. In this paper we compare these compactifications by explicit blow-ups and -downs when X is a projective homogeneous variety and d ≤ 3. Using the comparison result, we calculate the Betti numbers of the compactifications when X is a Grassmannian variety.
Abstract. We study the wall-crossing of the moduli spaces M α (d, 1) of α-stable pairs with linear Hilbert polynomial dm+ 1 on the projective plane P 2 as we alter the parameter α. When d is 4 or 5, at each wall, the moduli spaces are related by a smooth blow-up morphism followed by a smooth blow-down morphism, where one can describe the blow-up centers geometrically. As a byproduct, we obtain the Poincaré polynomials of the moduli spaces M(d, 1) of stable sheaves. We also discuss the wall-crossing when the number of stable components in Jordan-Hölder filtrations is three.1. Introduction 1.1. Motivation and Results. In moduli theory, for a given quasi-projective moduli space M 0 , various compactifications stem from the different view points for the moduli points of M 0 . After we obtain various compactified moduli spaces of M 0 , it is quite natural to ask the geometric relationship among them. Sometimes, this question is answered by birational morphisms between them, which enables us to obtain some geometric information (for example, the cohomology groups) of one space from that of the other [28,5].In this paper, we study the moduli space of semistable sheaves of dimension one on smooth projective surfaces [27], which recently gains interests in both mathematics and physics. This is an example of compactifications of the relative Jacobian variety, where we regard its general point as a sheafon a smooth curve C with pole along points p i of general position. In general, the moduli space of semistable sheaves is hard to study due to the lack of geometry of its boundary points. However, if n is equal to the genus of C, the sheaf F has a unique section up to scalar. So, we may alternatively consider the general point as a sheaf with a section, which in turn leads to another compactification, so called the moduli space of α-semistable pairs (more generally, the coherent systems [18]). When α is large, it can be shown that the moduli spaces of α-stable pairs are nothing but the relative 2010 Mathematics Subject Classification. 14D20.
We study the wall-crossing of the moduli spaces M α (d, 1) of α-stable pairs with linear Hilbert polynomial dm+ 1 on the projective plane P 2 as we alter the parameter α. When d is 4 or 5, at each wall, the moduli spaces are related by a smooth blow-up morphism followed by a smooth blow-down morphism, where one can describe the blow-up centers geometrically. As a byproduct, we obtain the Poincaré polynomials of the moduli spaces M(d, 1) of stable sheaves. We also discuss the wall-crossing when the number of stable components in Jordan-Hölder filtrations is three.2010 Mathematics Subject Classification. 14D20. Key words and phrases. Semistable pairs, Wall-crossing formulae, Blow-up/down, and Betti numbers.
Abstract. Let M d be the moduli space of stable sheaves on P 2 with Hilbert polynomial dm + 1. In this paper, we determine the effective and the nef cone of the space M d by natural geometric divisors. Main idea is to use the wall-crossing on the space of Bridgeland stability conditions and to compute the intersection numbers of divisors with curves by using the GrothendieckRiemann-Roch theorem. We also present the stable base locus decomposition of the space M 6 . As a byproduct, we obtain the Betti numbers of the moduli spaces, which confirm the prediction in physics.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.