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We give new contributions to the existence problem of canonical surfaces of high degree. We construct several families (indeed, connected components of the moduli space) of surfaces $S$ of general type with $p_g=5,6$ whose canonical map has image $Sigma$ of very high degree, $d=48$ for $p_g=5$, $d=56$ for $p_g=6$. And a connected component of the moduli space consisting of surfaces $S$ with $K^2_S = 40, p_g=4, q=0$ whose canonical map has always degree $geq 2$, and, for the general surface, of degree $2$ onto a canonical surface $Y$ with $K^2_Y = 12, p_g=4, q=0$. The surfaces we consider are SIP s, i.e. surfaces $S$ isogenous to a product of curves $(C_1 times C_2 )/ G$; in our examples the group $G$ is elementary abelian, $G = (mathbb{Z}/m)^k$. We also establish some basic results concerning the canonical maps of any surface isogenous to a product, basing on elementary representation theory.
In this paper we study emph{threefolds isogenous to a product of mixed type} i.e. quotients of a product of three compact Riemann surfaces $C_i$ of genus at least two by the action of a finite group $G$, which is free, but not diagonal. In particular
Let $S$ be a surface isogenous to a product of curves of unmixed type. After presenting several results useful to study the cohomology of $S$ we prove a structure theorem for the cohomology of regular surfaces isogenous to a product of unmixed type w
A complex surface $S$ is said to be isogenous to a product if $S$ is a quotient $S=(C_1 times C_2)/G$ where the $C_i$s are curves of genus at least two, and $G$ is a finite group acting freely on $C_1 times C_2$. In this paper we classify all regular
We consider a family of surfaces of general type $S$ with $K_S$ ample, having $K^2_S = 24, p_g (S) = 6, q(S)=0$. We prove that for these surfaces the canonical system is base point free and yields an embedding $Phi_1 : S rightarrow mathbb{P}^5$. Th
Using Bochner-Martinelli type residual currents we prove some generalizations of Jacobis Residue Formula, which allow proper polynomial maps to have `common zeroes at infinity, in projective or toric situations.