Canonical Surfaces an Hypersurfaces in Abelian Varieties

The present work deals with the canonical map of smooth, compact complex surfaces of general type, which induce a polarization of type (1,2,2) on an abelian threefold. The aim of the present study is to provide a geometric description of the canonical map of a smooth surface S of type (1,2,2) in an abelian threefold A in some special situations, and to prove that, when A and S are sufficiently general, the canonical system of S is very ample. It follows, in particular, a proof of the existence of canonical irregular surfaces in P5 with numerical invariants $p_g = 6$, $q = 3$ and $K^2 = 24$. This thesis is organized as follows: The first chapter deals with the basic theore- tical results concerning ample divisors on abelian varieties and their canonical map, which can be analytically represented in terms of theta functions. In this context, the example of surfaces in a polarization of type $(1, 1, 2)$ on an Abelian threefold, is of particular importance: the behavior of the canonical map of the pullback of a principal polarization by a degree 2 isogeny has been described by F. Catanese by investigating the canonical image and its defining projective equations by means of homological methods. In the last section of the first chapter, we treat in detail these results, as well as the connection with the analytical representation of the canonical map presented at the beginning of the same chapter. The polarization types $(1, 2, 2)$ and $(1, 1, 4)$ cannot be distinguished by considering only the numeric invariants of the ample surfaces in the respective linear systems. In the second chapter, we study the unramified bidouble covers of a smooth non-hyperelliptic curve of genus $3$, and we characterize the unramified bidouble covers of a general Jacobian $3$-folds, which carry a polarization of type $(1, 2, 2)$. In the third and last chapter of this thesis we investigate the behaviour of the canonical map of a general smooth surface in a polarization of type $(1,2,2)$ on an abelian threefold $A$, which is an etale quotient of a product of a $(2, 2)$-polarized abelian surface with a $(2)$-polarized elliptic curve. With this analysis and with some monodromy arguments, we prove the main result of this thesis, which states that the canonical system of a general smooth surface $S$ of type $(1, 2, 2)$ in a general abelian threefold $A$ yields a holomorphic embedding in $\mathbb{P}^5$.