Stability and applications (Q2032693)

In this well-written paper under review, the authors give a brief introduction to the theory of Bridgeland stability conditions, and then using the wall-crossing method, they review applications of this to a couple of important peoblems in algebraic geometry, namely (1) the Brill-Noether theorem, and (2) an upper bound on the genus of space curves. More precisely, for (1), after introducing stability conditions on \(K3\) surfaces, the authors outline the proof of the key theorem by \textit{A. Bayer} [Proc. Symp. Pure Math. 97, 3--27 (2018; Zbl 1451.14048)] which in turn implies the Lazarsfeld's version of Brill-Noether theorem for curves on \(K3\) surfaces (for the sake of simplicity, they consider a \(K3\) surface \(X\) with \(\mathrm{Pic}(X)=\mathbb{Z}\cdot H\) for some ample divisor \(H\)): Theorem 3.1. (Lazarsfeld). The Brill-Noether variety \(W^{r}_{d}(C)\) (i.e., the closed subset of \(\mathrm{Pic}_{d}(C)\) consisting of degree \(d\) line bundles \(L\) on a genus \(g\) curve \(C\), with \(h^{0}(L) \geq r+1\)) is non-empty if and only if \(\rho (r,d,g)\colon =g-(r+1)(g-d+1) \geq 0\). Moreover, in that case, \(W^{r}_{d}(C)\) has expected dimensin \(\mathrm{min}\{\rho(r,d,g),g\}\). Bayer reduces to looking at the subset \(V^{r}_{d}(C) \subset W^{r}_{d}(C)\) of constructible pure sheaves \(F\) in \(\mathrm{Coh}(X)\) supported on \(C\) with rank one, \(h^0(F)=r+1\), and \(\chi(F)=d+1-g\) to prove the following key theorem: Theorem 3.2. Assume \(0 < d \leq g-1\). The set \(V^{r}_{d}(|H|)\colon= \bigcup_{C\in |H|} V^{r}_{d}(C)\) is non-empty if and only if \(\rho (r,d,g)\geq 0\). Moreover, in that case there is a morphism \(V^{r}_{d}(|H|)\rightarrow M\), where \(M\) is is a non-empty open subset of a smooth projective irreducible holomorphic symplectic variety of dimension \(2\rho (r,d,g)\). Finally, each fiber is isomorphic to a Grassmannian variety of \((r + 1)\)-dimensional quotients of a vector space of dimension \(g - d + 2r + 1\). The authors give an overview of the proof of Theorem 3.2 using wall-crossing machinery: Starting from the large volume region, they describe the largest wall for objects with Mukai vector \((0,H,d+1-g)\) and characterize the objects in \(V^{r}_{d}(|H|)\) as the destabilizing locus at this wall (Lemma 3.9); then using an inequality for stable objects (Lemma 3.6) and further wall analysis, they complete the proof of Theorem 3.2. As for (2), after introducing tilt stability for \(\mathbb{P}^3\) (as an intermediate step to define Bridgeland stability conditions for threefolds due to \textit{A. Bayer} et al. [J. Algebr. Geom. 23, No. 1, 117--163 (2014; Zbl 1306.14005)], the authors outline their new proof [\textit{E. Macrì} and \textit{B. Schmidt}, Algebr. Geom. 7, No. 2, 153--191 (2020; Zbl 1455.14039)] for a more general version of Gruson-Peskine, Harris Theorem which gives an upper bound for the third chern character of the ideal sheaf of smooth space curves which are not contained in any surface of up to a certain degree; hence this gives and upper bound on the genus of these curves: Theorem 4.9. Let \(C \subset \mathbb{P}^3\) be a smooth curve of degree \(d\) and genus \(g\) that is not contained in a surface of degree \(l<k\). If \(d \geq k^2\), then \[2d+g-1=\mathrm{ch}_3(I_C) \leq \frac{d^2}{2k} + \frac{dk} {2}.\] As for the proof of this theorem, they start from the largest wall in the tilt-stability space and show that the only rank one destabilizing sub-objects of the desired ideal sheaves are line bundedles. Then they use a Bogomolove-Gieseker type inequality (Theorem 4.4, as suggested by \textit{A. Bayer} et al. [Invent. Math. 206, No. 3, 869--933 (2016; Zbl 1360.14057)] to find an upper bound on the third Chern character of rank zero objects which appear as destabilizing quotients of desired ideal sheaves. Using the fact that the desired upper bound in Theorem 3.2 is a decreasing function in \(k\), when \(d \geq k^2\) (Lemma 4.8), they complete the proof.