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Register a new userOn the canonical map of some surfaces isogenous to a product
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Authors
Fabrizio Catanese
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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.
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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, we are interested in the systematic construction and classification of these varieties. Our main result is the full classification of threefolds isogenous to a product of mixed type with $chi(mathcal O_X)=-1$ assuming that any automorphism in $G$, which restricts to the trivial element in $Aut(C_i)$ for some $C_i$, is the identity on the product. Since the holomorphic Euler-Poincare-characteristic of a smooth threefold of general type with ample canonical class is always negative, these examples lie on the boundary, in the sense of threefold geography. To achieve our result we use techniques from computational group theory. Indeed, we develop a MAGMA algorithm to classify these threefolds for any given value of $chi(mathcal O_X)$.
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 with $chi (mathcal{O}_S)=2$. In particular we found two families of surfaces of general type with maximal Picard number.
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 surfaces isogenous to a product with $chi(mathcal O_S) = 2$ under the assumption that the action of $G$ is unmixed i.e. no element of $G$ exchange the factors of the product $C_1 times C_2$.
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 $Phi_1 : S rightarrow mathbb{P}^5$. This result answers a question posed by G. and M. Kapustka. We discuss some related open problems, concerning also the case $p_g(S) = 5$, where one requires the canonical map to be birational onto its image.
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.
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