Computing the Walls Associated to Bridgeland Stability Conditions on Projective Surfaces

By Antony Maciocia. 2012

114 citations recorded by the source corpus.

We derive constraints on the existence of walls for Bridgeland stability conditions for general projective surfaces. We show that in suitable planes of stability conditions the walls are bounded and derive conditions for when the number of walls is globally finite. In examples, we show how to use the explicit conditions to locate walls and sometimes to show that there are no walls at all.

Subfields: Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/5e84c6adeadcd52bb8a3df8a486006f151f4a283

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In Geometry and Topology

In Statistical and Nonlinear Physics

Citations in the corpus

The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds (2014, background): “analogous to the case of walls for Bridgeland stability on surfaces, treatedmost systematically in [25].”
A generalized Bogomolov–Gieseker inequality for the three-dimensional projective space (2012, methodology): “If ch0(A)≥ 3, then by using again [Maciocia 2012], we can deduce that BA is contained in the interior of B2.”
Stability conditions and extremal contractions (2012, background): “moduli spaces of objects whose numerical class is equal to that of a skyscraper sheaf. The Bridgeland stability conditions on arbitrary surfaces are constructed in [1]. In the works [27], [28], [35], [24], [25], the structure of walls and wall-crossing phenomena with respect to these stability condi”
Bridgeland Stability Conditions on Fano Threefolds (2016, background): “f the set of semistable objects with class vchanges at S. The structure of walls in tilt stability is rather simple. Part (1) - (5) is usually called Bertram’s Nested Wall Theorem and appeared rst in [Mac14a], while part (6), (7), and (8) are in Lemma 2.7 and Appendix A of [BMS16]. The last part of ”
Projectivity and birational geometry of Bridgeland moduli spaces (2012, unknown): “The analog of Theorem 1.3 for abelian surfaces has been proved in [66]; a diﬀerent method to prove projectivity was established in [53,54].”
Projectivity and birational geometry of Bridgeland moduli spaces on an Enriques surface (2014, methodology): “Following [35], we call the codimension-1 submanifolds from Proposition 4.”
The Minimal Model Program for the Hilbert Scheme of Points on P^2 and Bridgeland Stability (2012, background): “af of ﬁnite length. (c) Torsion-free sheaves in Qs that are not Mumford semistable are not (s,t)-semistable for large t. (d) Line bundles OP2(k), k>s, are stable objects of As for all t. 1Maciocia [Ma] recently proved that Bridgeland walls for smooth projective surfaces of Picard rank one are nes”
Lectures on Bridgeland Stability (2016, result): “It was originally observed by Bertram and completely proved in [Mac14b].”