Stability of rank-3 Lazarsfeld-Mukai bundles on \(K3\) surfaces (Q2843977)

Let \(S\) be a \(K3\) surface and \(C\subset S\) a smooth connected curve of genus \(g\). If \(A\) is a degree \(d\) line bundle on \(C\) having \(r+1\) linearly independent sections and hence can be considered as a globally generated sheaf on \(S\), the associated Lazarsfeld--Mukai (LM) bundle is defined to be the dual of the kernel of the canonical evaluation map \(H^0(C,A)\otimes \mathcal{O}_S\to A\). These bundles are connected to Brill--Noether theory on \(C\). Given an ample line bundle \(L\), the paper under review studies the \(\mu_L\)-stability (which is the usual slope-stability with respect to \(L\)) of LM-bundles associated with line bundles on curves in the linear system defined by \(L\).NEWLINENEWLINEThe author gives some background information concerning LM-bundles and Mumford stability for sheaves on \(K3\) surfaces in Section 2 and 3, respectively. In Section 4 the following statement is proved. Let \(S\) be a \(K3\) surface and \(L\) an ample line bundle such that a general smooth connected curve in \(|L|\) has genus \(g\), Clifford dimension \(1\) and maximal gonality. If the Brill--Noether number \(\rho(g,1,d)\) is positive, then the LM-bundle associated with a general complete base point free \(g^1_d\) on \(C\) (roughly, this is a pair consisting of a degree \(d\) line bundle on \(C\) and \(2\) global sections) is \(\mu_L\)-stable. This gives a new proof of a result on some geometric aspects of certain families of Brill--Noether varieties (previously established by Aprodu and Farkas) described in Theorem 4.3.NEWLINENEWLINEStarting in Section 5, the stability of LM-bundles with a complete base point free \(g^2_d\) is considered. These bundles are now of rank \(3\), have \(c_2=d\) and \(\det=L\), and the author studies the stability of these bundles. There are several cases to be distinguished: The bundle can be unstable and then we look at its Harder--Narasimhan filtration, or the bundle is properly semistable and in this case its Jordan--Hölder filtration is of importance. These two cases can be further subdivided into subcases which depend, for example, on the rank of some factors in the filtrations. All this is done in Sections 5-8 using, in particular, stacks of filtrations, and once the investigation is finished, the author can prove Theorems 1.1 and 1.2, where the first one gives, under some conditions, the existence of a line bundle on \(S\) adapted to \(|L|\), while the second one studies the variety \(\mathcal{W}^2_d(|L|)\) over the locus of smooth and connected curves in \(|L|\) (roughly, the fibre of \(\mathcal{W}^2_d(|L|)\) over a curve \(C\) is the Brill--Noether variety \(W^2_d(C)\)).NEWLINENEWLINEIn Section 9 an application towards transversality of Brill--Noether and Gieseker--Petri loci is presented (the latter consists of those curves of genus \(g\) which violate the Gieseker--Petri theorem). Finally, the last section gives an application to higher rank Brill--Noether theory.