The Zak Map of a Curve. Gaussian Maps

Abstract

We first notice that the condition ${H^1}\left( {{T_Y} \otimes {L^{ - 1}}} \right) = 0$ of Corollary 3.3 is never fulfilled for any smooth projective curve Y embedded in ℙ _n such that Y is non-degenerate and of codimension ≥ 2. The aim of this chapter is to provide an interpretation of the Zak map of a linearly normal smooth curve Y ⊂ ℙ _n in terms of the so-called Gaussian maps. The advantage of Gaussian maps comes from the fact that in certain cases there are methods to check their surjectivity, see e. g. [147], [148], [38] and the references therein.