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Accepted Manuscript Izzet Coskun, Donghoon Hyeon, Junyoung Park Castelnuovo-Mumford regularity and Bridgeland stability of points in the projective plane Proceedings of the American Mathematical Society DOI: 10.1090/proc/13470 Accepted Manuscript This is a preliminary PDF of the author-produced manuscript that has been peer-reviewed and accepted for publication. It has not been copyedited, proofread, or finalized by AMS Production staff. Once the accepted manuscript has been copyedited, proofread, and finalized by AMS Production staff, the article will be published in electronic form as a \Recently Published Article" before being placed in an issue. That electronically published article will become the Version of Record. This preliminary version is available to AMS members prior to publication of the Version of Record, and in limited cases it is also made accessible to everyone one year after the publication date of the Version of Record. The Version of Record is accessible to everyone five years after publication in an issue. CASTELNUOVO-MUMFORD REGULARITY AND BRIDGELAND STABILITY OF POINTS IN THE PROJECTIVE PLANE IZZET COSKUN, DONGHOON HYEON, AND JUNYOUNG PARK Abstract. In this paper, we study the relation between Castelnuovo-Mumford 2 regularity and Bridgeland stability for the Hilbert scheme of n points on P . n For the largest b 2 c 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. 1. Introduction In this paper, we consider the relation between the Castelnuovo-Mumford reg- 2 ularity and the Bridgeland stability of zero-dimensional subschemes of P . Our study is motivated by the following result which relates geometric invariant theory (GIT) stability and Castelnuovo-Mumford regularity. 3g-4 Theorem. [HH13, Corollary 4.5] Let C ⊂ P be a c-semistable bicanonical curve. Then OC is 2-regular. Note that c-semistability of curves [HH13, Definition 2.6] is a purely geomet- ric notion concerning singularities and subcurves, whereas Castelnuovo-Mumford regularity is an algebraic notion regarding the syzygies of ideal sheaves. 2 2 For points in P , a similar but weaker statement holds. A set of n points in P is GIT semistable if and only if at most 2n=3 of the points are collinear, in which case the regularity is at most 2n=3. However, the regularities of semistable points cover a broad spectrum. Our goal in this paper is to use Bridgeland stability to obtain a closer relationship between stability and regularity. There is a distinguished half-plane H = f(s; t)js > 0; t 2 Rg of Bridgeland stability 2 conditions for P . Let ξ be a Chern character. The half-plane H admits a wall-and- chamber decomposition, where in each chamber the set of Bridgeland semistable objects with Chern character ξ remains constant. The Bridgeland walls where an ideal sheaf of points is destabilized consist of the vertical line s = 0 and a finite set of nested semicircular walls Wc centered along the Date: September 3, 2016. 2010 Mathematics Subject Classification. 14C05, 13D02, 14D20 primary, 13D99, 14D99, 14C99 secondary. Key words and phrases. Castelnuovo-Mumford regularity, Hilbert schemes of points, Bridge- land stability, monomial schemes. The first author was partially supported by the NSF CAREER grant DMS-0950951535 and the NSF grant DMS-1500031; the second author was supported by the following grants funded by the government of Korea: NRF grant 2011-0030044 (SRC-GAIA) and NRF grant NRF- 2013R1A1A2010649. 1 This is a pre-publication version of this article, which may differ from the final published version. CopyrightSep 3 restrictions2016 09:35:20 may apply. EDT Version 2 - Submitted to PROC 2 IZZET COSKUN, DONGHOON HYEON, AND JUNYOUNG PARK 3 s-axis at s = -c - 2 < 0 [ABCH13, Section 6]. Since the semicircular Bridgeland walls are nested, we can order them by inclusion. If an ideal sheaf IZ is destabilized along the wall Wc, then IZ is Bridgeland stable in the region bounded by Wc and s = 0. Let σ ≺ σ 0 if all σ 0-semistable ideal sheaves with Chern character ξ are 2[n] σ-semistable. Consequently, Bridgeland stability induces a stratification of P 2[n] α P = X ; α a where α 2[n] X = fZ 2 P j IZ is α-semistable but β-unstable 8α ≺ βg and α runs over a Bridgeland stability condition in each chamber. We have α S β X = βα X (see Section 2). By [ABCH13, Sections 9,10] and [LZ], this strat- 2[n] ification coincides with the stratification of P according to the stable base loci of linear systems. Recall that the effective cone of a variety has a wall and cham- ber decomposition such that in each chamber the stable base locus of the divisors remain constant. Similarly, there is a stratification induced by Castelnuovo-Mumford regularity: 2[n] r-reg P = X ; r2 aZ where Xr-reg is the collection of ideals whose Castelnuovo-Mumford regularity is r. The regularity, being a cohomological invariant [Eis95, Proposition 20.16], is r-reg r 0-reg upper-semicontinuous and we have X = r 0≥r X . This naturally raises the question of comparing` the two stratifications. We will n show that a general scheme destabilized at one of the b 2 c largest Bridgeland walls has smaller regularity than the general scheme destabilized along the larger walls. Our main theorem will be proved in Section 5: 2 Theorem. Let pi be the maximal ideal of the closed point pi 2 P , i = 1; : : : ; s. s mi Let Z be the subscheme given by \i=1pi and let n be its length. Define t h := max mi pi ; : : : ; pit are collinear : 8 j 1 9 j=1 <X = If n ≤ 2h - 3, then Z is destabilized at the wall W . In particular, general : reg(Z)-1 ; points destabilized at Wk+1 have higher regularity than those destabilized at Wk, n 8k ≥ 2 - 1. For zero-dimensional subschemes cut out by monomials, we have a more precise connection between regularity and Bridgeland stability: 2 Proposition. Let Z be a zero-dimensional monomial scheme in P . If the ideal 3 sheaf IZ is destabilized at the wall Wµ(Z) with center x = -µ(Z)- 2 , then 3 (reg(I )- 1) ≤ µ(Z) ≤ reg(I )- 1: 4 Z Z (1) The left equality holds if and only if reg(IZ) + 1 = 2m is even and IZ = hxm; ymi a a b br (2) The right equality holds if and only if IZ = hx 1 ; x 2 y 2 ; : : : ; y i with max1≤i≤r-1(ai + bi+1 - 1) ≤ max(a1; br). This is a pre-publication version of this article, which may differ from the final published version. CopyrightSep 3 restrictions2016 09:35:20 may apply. EDT Version 2 - Submitted to PROC REGULARITY AND BRIDGELAND STABILITY 3 In particular, for Borel fixed ideals, the regularity and the Bridgeland stability completely determine each other: 2 Corollary. Let Z ⊂ P be a zero-dimensional monomial scheme whose ideal is Borel-fixed (which holds if it is a generic initial ideal, for instance). Then the ideal sheaf IZ is destabilized at the wall Wreg(IZ)-1. In general, the relation between regularity and the Bridgeland slope is not mono- tonic. Let Z1 and Z2 be two schemes of length n destabilized along Wµ(Z1) and Wµ(Z2), respectively. It may happen that while reg(Z1) > reg(Z2), we have µ(Z1) < µ(Z2). We close the introduction with the following simple but illustrative example. 4 4 Example 1.1. Let Z1 and Z2 be the monomial scheme defined by hx ; y i and hx6; x5y; x4y2; xy3; y4i, respectively. Both are of length 16, and by the arguments 9 of Section 3, we see that reg(IZ1 ) = 7; reg(IZ2 ) = 6 and µ(Z1) = 2 ; µ(Z2) = 5. We work over an algebraically closed field K of characteristic zero. Acknowledgements. We would like to thank Aaron Bertram and Jack Huizenga for enlightening conversations. 2. Preliminaries on Bridgeland stability conditions 2 We briefly review the basics of Bridgeland stability conditions on P . We refer b 2 the reader to [ABCH13] and [CH14] for more details. Let D (P ) be the bounded 2 2 b 2 derived category of coherent sheaves on P , and K(P ) be the K-group of D (P ). 2 Definition 2.1. A Bridgeland stability condition on P consists of a pair (A; Z), b 2 2 where A is the heart of a t-structure on D (P ) and Z : K(P ) C is a homomor- phism (called the central charge) satisfying • if 0 6= E 2 A, Z(E) lies in the semi-closed upper half-plane! freiπθ j r > 0; 0 < θ ≤ 1g. • (A; Z) has the Harder-Narasimhan property, which will be defined below. Definition 2.2. Writing Z = -d + ir, the slope µ(E) of 0 6= E 2 A is defined by µ(E) = d(E)=r(E) if r(E) 6= 0 and µ(E) = otherwise. Definition 2.3. An object E 2 A is called stable (resp. semistable) if for every proper subobject F ⊂ E in A, µ(F) < µ(E)1(resp. µ(F) ≤ µ(E)). Definition 2.4. The pair (A; Z) has the Harder-Narasimhan property if any nonzero object E 2 A admits a finite filtration 0 ⊂ E0 ⊂ E1 ⊂ · · · ⊂ En = E such that each Harder-Narasimhan factor Fi = Ei=Ei-1 is semistable and µ(F1) > µ(F2) > ··· > µ(Fn).
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