On the surjectivity of weighted Gaussian maps

TL;DR

This paper investigates the surjectivity of weighted Gaussian maps, generalizing standard Gaussian maps, to understand the local geometry of curves with higher roots of the canonical bundle, providing bounds and specific locus descriptions.

Contribution

It introduces a generalized class of Gaussian maps, establishes bounds on the Zariski tangent space dimension, and describes the locus for complete intersection curves.

Findings

Bound on the Zariski tangent space dimension, sharp for r=0

Description of the locus for complete intersection curves

Generalization of Gaussian maps to weighted versions

Abstract

We study the surjectivity of suitable weighted Gaussian maps which provide a natural generalization of the standard Gaussian maps and encode the local geometry of the locus of curves endowed with a higher root of the canonical bundle having associated linear system of dimension at least r + 1. In particular, we get a bound on the dimension of its Zariski tangent space, which turns out to be sharp in the special case r = 0. Finally, we describe this locus in the case of complete intersection curves.