The rank of the 2nd Gaussian map for general curves

TL;DR

This paper determines the injectivity and surjectivity of the 2nd Gaussian map for general curves of genus g, showing a transition at g=17 to 18, based on degeneration techniques.

Contribution

It establishes the precise genus bounds for the 2nd Gaussian map's injectivity and surjectivity for general curves, using degeneration to binary curves.

Findings

Injective for g <= 17

Surjective for g >= 18

Uses degeneration to binary curves

Abstract

We prove that, for the general curve of genus g, the 2nd Gaussian map is injective if g <= 17 and surjective if g >= 18. The proof relies on the study of the limit of the 2nd Gaussian map when the general curve of genus g degenerates to a general stable binary curve, i.e. the union of two rational curves meeting at g+1 points.