Very ampleness of the canonical bundle of surfaces of type (1, 2, 2) on abelian threefolds

Abstract: The present work deals with the canonical map of smooth, compact complex surfaces of general type in a polarization of type $(1,2,2)$ on an abelian threefold. A natural and classical question is whether the canonical system of such surfaces is very ample in the general case. In this work, we provide a positive answer to this question. First, we describe the structure of the canonical map of those smooth ample surfaces of type $(1,2,2)$ in an abelian threefold which are bidouble cover of principal polarizations. Then, we study the general behavior of the canonical map of general ample surfaces $\mathcal{S}$, yielding a $(1,2,2)$-polarization on an abelian threefold $A$ which is isogenous to a product. By combining these descriptions, we show that the canonical map yields a holomorphic embedding when $A$ and $\mathcal{S}$ are both sufficiently general. It follows, in particular, a proof of the existence of canonical irregular surfaces in $\mathbb{P}^5$ with numerical invariants $p_g = 6$, $q = 3$ and $K^2 = 24$.