The Minimal Model Program for the Hilbert Scheme of Points on P^2 and Bridgeland Stability

By D. Arcara, Aaron Bertram, Izzet Coskun, Jack Huizenga. 2012

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In this paper, we study the birational geometry of the Hilbert scheme of n points on P^2. We discuss the stable base locus decomposition of the effective cone and the corresponding birational models. We give modular interpretations to the models in terms of moduli spaces of Bridgeland semi-stable objects. We construct these moduli spaces as moduli spaces of quiver representations using G.I.T. and thus show that they are projective. There is a precise correspondence between wall-crossings in the Bridgeland stability manifold and wall-crossings between Mori cones. For n at most 9, we explicitly determine the walls in both interpretations and describe the corresponding flips and divisorial contractions.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e5091f3eeaaf3b371d6669fac6cb85b6c5a70149

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Citations in the corpus

Lectures on Bridgeland Stability (2016, background): “24 has an affirmative complete answer only for the projective plane P2 (see [ABCH13]), P1 × P1 and the blow up of a point in P2 (see [AM15]), and partial answers for other surfaces, including abelian surfaces (see [MYY14, YY14]), K3 surfaces (see [BM14a]), and Enriques surfaces (see [Nue16, Yos16]).”
A generalized Bogomolov–Gieseker inequality for the three-dimensional projective space (2012, background): “It would be very interesting to study how moduli spaces of Bridgeland semistable objects vary when varying the stability condition (very much like the situation described in [Arcara et al. 2013; Maciocia and Meachan 2013; Lo and Qin 2011; Minamide et al. 2011; Yanagida and Yoshioka 2012; Bayer and M”
A generalized Bogomolov–Gieseker inequality for the smooth quadric threefold (2013, background): “Since C is the category of representations of a quiver with relations with simple objects O(−1)[3], S(−1)[2],O[1],O(1), we can interpret v(F ) = (a, b, c, d) as the dimension vector of that representation.”
Projectivity and birational geometry of Bridgeland moduli spaces on an Enriques surface (2014, methodology): “I. Coskun and J. Huizenga, ‘The nef cone of the moduli space of sheaves and strong Bogomolov inequalities’, Preprint, 2015, arXiv:math/1512.02661.”
Zariski decompositions of numerical cycle classes (2013, methodology): “ decompositions whose positive parts are not nef. 7.2. Cycles on P2[2]. In order to cohere to the cited references, in this section we will consider projective spaces of lines instead of hyperplanes. [ABCH13] gives an extensive analysis of the divisor theory for Hilbert schemes of points on P2. It w”
Birational models of moduli spaces of coherent sheaves on the projective plane (2016, background): “This is first proved by Coskun, Huizenga and Woolf.”
The effective cone of the moduli space of sheaves on the plane (2014, background): “by the exceptional divisor Bof the HilbertChow morphism P2[n] → P2(n). The secondary edge of the movable cone is spanned by the divisor Hgiven as the locus of schemes meeting a ﬁxed line in the plane [ABCH]. • If r(ξ) = 2, the exceptional locus Eof the morphism M(ξ) → MDUY(ξ) is divisorial, and it s”
Projectivity and birational geometry of Bridgeland moduli spaces (2012, unknown): “Our construction was directly inspired by the results of [4].”