Lifts of line bundles on curves on \(K3\) surfaces (Q6564798)

The paper under review is devoted to two tasks. In the first part, they consider a classical conjecture that the existence of a Brill-Neother special curve should induce the existence of a Brill-Neother special polarized \(K3\) surface. The second part is dedicated to a study of the structure of the linear system that determines a double covering of a plane curve branching at some sextic curve, namely, when the double covering curve is sitting in a \(K3\) surface of degree \(2\).\N\NIn part one, consider a curve \(C\) and a line bundle \(A\) on \(C\) such that the genus \(g\) of \(C\), the degree \(d\) of \(A\), and the dimension of the linear system \(|A|\) have appropriate conditions so that the Brill-Neother number \(\rho(g, d, r)\) is negative, namely, the curve \(C\) is Brill-Neother special. The authors proved the main theorem (Theorem 1.1) which states that if \(C\) is on a \(K3\) surface \(X\), and \(A\) computes the Clifford index of \(C\), then, there exists a Donagi-Morrison lift. In fact, in this context, the curve \(C\) is regarded as a smooth member of the linear system \(|L|\) of a not-necessarily ample line bundle \(L\) of \(X\).\N\NIn the second part, they turn to their attention to the properties of a possible curve \(C\) in a \(K3\) surface which is obtained as a double covering of the projective plane curve of degree \(k\). They prove such a curve \(C\) should have genus \(\leq k^2+1\). In particular, if there exists a gonality pencil on \(C\) which has no lift and the genus of \(C\) attains the maximum, then, it is possible that \(C\) is linearly equivalent to a multiple of a curve \(B\) of genus 2 such that the corresponding sheaf \(\mathcal{O}_X(B)\) is a lift of a base-point free net of degree \(2k\) associated with a morphism from \(C\) to a projective plane.\N\NThe proofs are due to their high-skilled techniques to manipulate exact sequences that involve the (generalized) Lazarsfeld-Mukai bundles on \(K3\) surfaces, as well as authentic theory of algebraic curves.