Higher Gaussian maps for special classes of curves

The main task of the thesis is the computation of the rank of higher Gaussian maps for special classes of smooth projective curves. First, we study even order higher Gaussian maps of the canonical bundle on hyperelliptic curves and we determine their rank, giving explicit descriptions of their kernels. Then we use this descriptions to investigate the hyperelliptic Torelli map $ and its second fundamental form. We study isotropic subspaces of the tangent space to the moduli space of hyperelliptic curves of genus g at a point [C], with respect to its second fundamental form. In particular, for any Weierstrass point p of the curve C, we construct a space generated by higher Schiffer variations at p, such that the only isotropic tangent direction is the standard Schiffer variation at the Weierstrass point p. In a similar way as in the hyperelliptic case, we study even order Gaussian maps on particular trigonal curves, namely some specific 3:1 cyclic covers of the projective line. We manage to compute a lower bound for the rank of the even order Gaussian maps on these curves, and hence on the general trigonal curve. Then we use this computations to show the non existence of extra asymptotic direction for these kinds of trigonal curves in suitable spaces. We mention that in both hyperelliptic and trigonal case, the techniques that we use are based on explicit computations. Using very different techniques, such as cohomological tools, we then study higher Gaussian (or Wahl) maps for the canonical bundle of certain smooth projective curves. More precisely, we determine the rank of higher Gaussian maps of the canonical bundle for plane curves and for curves contained in a sufficiently ample line bundle on an Enriques surface. For plane curves, we find out that these maps are never surjective, and that the corank is independent of the degree of the curve. For curves on Enriques surfaces we find sufficient conditions for their surjectivity.