On the splitting of Lazarsfeld-Mukai bundles on \(K3\) surfaces. II.
Lazarsfeld-Mukai bundles form a very important class of vector bundles on \(K3\) surfaces. Concretely, given \(C\subset X\) a smooth curve in a \(K3\) surface, such that \(\mathcal O_X(C)\) is globally generated, and a base point free divisor \(Z\) on \(C\), the associated Lazarsfeld-Mukai bundle \(E_{C,Z}:=F_{C,Z}^\vee\) is given by the dual of the kernel of the evaluation map \[ F_{C,Z}:=\ker\Bigl(\operatorname{ev}: H^0(\mathcal O_C(Z))\otimes \mathcal O_X\to \mathcal O_C(Z)\Bigr)\,. \] In his previous article [J. Algebra 447, 445--454 (2016; Zbl 1356.14029)], the author studied, under the additional assumption that \(H:=\mathcal O_X(C)\) is very ample, the question whether a simple Lazarsfeld-Mukai bundle \(E_{C,Z}\) of rank 2 can be written as an extension of two line bundles. Since \(c_1(E_{C,Z})=[C]\), such an extension must be of the form \[ 0\to H\otimes L^\vee\to E_{C,Z}\to L\to 0\,. \] Proposition 3.1 of [loc. cit.] stated that, in the case that \(X\) is a quartic hypersurface and \(H^1(H\otimes L^\vee)=0\), the intersection numbers must be given by \(L. H=5\) and \(L^2=2\). This statement was wrong, and the main purpose of the present article is to correct it. Namely, only the weaker statement that \((L. H,L^2)\in\{(5,2),(3,0),(4,0)\}\) is true. After two preliminary sections, the author proves the corrected weaker statement in Section 4. In Sections 5 and 6, it is shown that the case \((L. H,L^2)=(3,0)\) really occurs. This shows that not only the proof of the older statement, but also the statement itself, was incorrect. Whether or not the case \((L. H,L^2)=(4,0)\) really occurs remains open.
scientific article; zbMATH DE number 6990640
scientific article