Abstract
In this paper, we study the relation between Castelnuovo-Mumford regularity and Bridgeland stability for the Hilbert scheme of n n points on P 2 \mathbb {P}^2 . For the largest ⌊ n 2 ⌋ \lfloor \frac {n}{2} \rfloor Bridgeland walls, we show that the general ideal sheaf destabilized along a smaller Bridgeland wall has smaller regularity than one destabilized along a larger Bridgeland wall. We give a detailed analysis of the case of monomial schemes and obtain a precise relation between the regularity and the Bridgeland stability for the case of Borel fixed ideals.
Highlights
We consider the relation between the Castelnuovo-Mumford regularity and the Bridgeland stability of zero-dimensional subschemes of P2
Our study is motivated by the following result which relates geometric invariant theory (GIT) stability and Castelnuovo-Mumford regularity
Note that c-semistability of curves [HH13, Definition 2.6] is a purely geometric notion concerning singularities and subcurves, whereas Castelnuovo-Mumford regularity is an algebraic notion regarding the syzygies of ideal sheaves
Summary
We consider the relation between the Castelnuovo-Mumford regularity and the Bridgeland stability of zero-dimensional subschemes of P2. The half-plane H admits a wall-andchamber decomposition, where in each chamber the set of Bridgeland semistable objects with Chern character ξ remains constant. Castelnuovo-Mumford regularity, Hilbert schemes of points, Bridgeland stability, monomial schemes. If an ideal sheaf IZ is destabilized along the wall Wc, IZ is Bridgeland stable in the region bounded by Wc and s = 0. For zero-dimensional subschemes cut out by monomials, we have a more precise connection between regularity and Bridgeland stability: Proposition. For Borel fixed ideals, the regularity and the Bridgeland stability completely determine each other: Corollary. Let Z ⊂ P2 be a zero-dimensional monomial scheme whose ideal is Borel-fixed (which holds if it is a generic initial ideal, for instance). We work over an algebraically closed field K of characteristic zero
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