Globally generated vector bundles on P3 and the unirationality of Mg, g≤13

Abstract

Chang and Ran proved, in 1984, the unirationality of the moduli spaces of curves of genus 11, 12, and 13. They consider a family U of nonsingular space curves. If U is non-empty, it dominates the moduli space. Considering certain vector bundles related to the curves, one shows that U is unirational. Finally, Chang and Ran prove that U is non-empty by constructing smoothable reducible curves. We prove, instead, the existence of (almost) globally generated vector bundles of the above kind. The dependency locus of (rank − 1) general global sections of such a bundle is a nonsingular curve from U.

Key Words:

• Vector bundles on projective spaces
• moduli spaces of curves
• unirationality
• space curves

2020 Mathematics Subject Classification:

• Primary: 14J60
• Secondary: 14H10
• 14H50

Acknowledgements

N. Manolache expresses his thanks to the Institute of Mathematics, Oldenburg University, especially to Udo Vetter, for warm hospitality during the preparation of this work. We are, also, grateful to the referee for many detailed suggestions that helped us improve the presentation of the paper.