On the surjectivity of weighted Gaussian maps

Abstract

We study the surjectivity of suitable weighted Gaussian maps \(\gamma _{a,b}(X,L)\) which provide a natural generalization of the standard Gaussian maps and encode the local geometry of the locus \(\mathfrak{Th }^r_{g,h}\subset \mathcal M _g\) of curves endowed with an \(h\)-th root \(L\) of the canonical bundle satisfying \(h^0(L)\ge r+1\). In particular, we get a bound on the dimension of its Zariski tangent space, hich turns out to be sharp in the special case \(r=0\). Finally, we describe this locus in the case of complete intersection curves.