Abstract
We explicitly construct Brill–Noether general K3 surfaces of genus 4, 6 and 8 having the maximal number of elliptic pencils of degrees 3, 4 and 5, respectively, and study their moduli spaces and moduli maps to the moduli space of curves. As an application we prove the existence of Brill–Noether general K3 surfaces of genus 4 and 6 without stable Lazarsfeld–Mukai bundles of minimal c_2.
Highlights
It is well-known that a general curve of genus g ≤ 9 or g = 11 can be realized as a linear section of a primitively polarized K 3 surface, cf. [26,28]
The Lazarsfeld–Mukai bundle associated to a pencil on a smooth curve on the K 3 surface induced by an elliptic pencil on the surface is necessarily not stable, cf
Section 3: We prove that a general curve C of genus 4 is a linear section of a smooth K 3 surface S such that its two g31s are induced by two elliptic pencils |E1| and |E2| on S satisfying C ∼ E1 + E2, cf
Summary
It is well-known that a general curve of genus g ≤ 9 or g = 11 can be realized as a linear section of a primitively polarized K 3 surface, cf [26,28]. Using Mukai’s results, we will study projective models of Brill–Noether general K 3 surfaces of genus g ∈ {4, 6, 8} containing the maximal possible number of elliptic pencils of degree g 2. Section 3: We prove that a general curve C of genus 4 is a linear section of a smooth K 3 surface S such that its two g31s (which are well-known to be auto-residual) are induced by two elliptic pencils |E1| and |E2| on S satisfying C ∼ E1 + E2, cf Proposition 3.4.
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