arXiv subjects
Publications and source records attributed to Carlos Rito.
We give a list of possibilities for surfaces of general type with $p_g=0$ having an involution $i$ such that the bicanonical map of $S$ is not composed with $i$ and $S/i$ is not rational. Some examples with $K^2=4,...,7$ are constructed as double coverings of an Enriques surface. These surfaces have a description as bidouble coverings of the plane.
Let $S$ be a smooth minimal surface of general type with a (rational) pencil of hyperelliptic curves of minimal genus $g$. We prove that if $K_S^2<4χ(\mathcal O_S)-6,$ then $g$ is bounded. The surface $S$ is determined by the branch locus of the covering $S\rightarrow S/i,$ where $i$ is the hyperelliptic involution of $S.$ For $K_S^2<3χ(\mathcal O_S)-6,$ we show how to determine the possibilities for this branch curve. As an application, given $g>4$ and $K_S^2-3χ(\mathcal O_S)<-6,$ we compute the maximum value for $χ(\mathcal O_S).$ This list of possibilities is sharp.
We study surfaces of general type $S$ with $p_g=0$ and $K^2=3$ having an involution $i$ such that the bicanonical map of $S$ is not composed with $i$. It is shown that, if $S/i$ is not rational, then $S/i$ is birational to an Enriques surface or it has Kodaira dimension $1$ and the possibilities for the ramification divisor of the covering map $S\rightarrow S/i$ are described. We also show that these two cases do occur, providing an example. In this example $S$ has a hyperelliptic fibration of genus $3$ and the bicanonical map of $S$ is of degree $2$ onto a rational surface.
Two Magma functions are given: one computes linear systems of plane curves with non-ordinary singularities and the other computes a scheme which parametrizes given degree plane curves with given singularities. These functions provide an efficient tool to construct explicit equations of singular plane algebraic curves. By computing singular branch curves, we obtain equations of normal quartic surfaces in $\mathbb C\mathbb P^3$ having the following combinations of rational double points: $\mathsf D_5\mathsf E_7\mathsf E_7,$ $\mathsf D_7\mathsf D_6\mathsf D_6,$ $\mathsf E_6\mathsf D_8\mathsf D_5,$ $\mathsf E_6\mathsf D_{13},$ $\mathsf E_6\mathsf E_6\mathsf D_7,$ $\mathsf E_6\mathsf E_8\mathsf D_5,$ $\mathsf E_7\mathsf D_6\mathsf D_6,$ $\mathsf E_7\mathsf D_{12},$ $\mathsf E_7\mathsf E_6\mathsf D_6.$ These are all possible cases with total Milnor number 19 which have no point of type $\mathsf A_n.$
In this paper some numerical restrictions for surfaces with an involution are obtained. These formulas are used to study surfaces of general type $S$ with $p_g=q=1$ having an involution $i$ such that $S/i$ is a non-ruled surface and such that the bicanonical map of $S$ is not composed with $i.$ A complete list of possibilities is given and several new examples are constructed, as bidouble covers of surfaces. In particular the first example of a minimal surface of general type with $p_g=q=1$ and $K^2=7$ having birational bicanonical map is obtained.
Let $S$ be a {\em Todorov surface}, {\it i.e.}, a minimal smooth surface of general type with $q=0$ and $p_g=1$ having an involution $i$ such that $S/i$ is birational to a $K3$ surface and such that the bicanonical map of $S$ is composed with $i.$ The main result of this paper is that, if $P$ is the minimal smooth model of $S/i,$ then $P$ is the minimal desingularization of a double cover of $\mathbb P^2$ ramified over two cubics. Furthermore it is also shown that, given a Todorov surface $S$, it is possible to construct Todorov surfaces $S_j$ with $K^2=1,...,K_S^2-1$ and such that $P$ is also the smooth minimal model of $S_j/i_j,$ where $i_j$ is the involution of $S_j.$ Some examples are also given, namely an example different from the examples presented by Todorov in \cite{To2}.
This paper describes how to compute equations of plane models of minimal Du Val double planes of general type with $p_g=q=1$ and $K^2=2,...,8.$ A double plane with $K^2=8$ having bicanonical map not composed with the associated involution is also constructed. The computations are done using the algebra system Magma.
This paper classifies surfaces of general type $S$ with $p_g=q=1$ having an involution $i$ such that $S/i$ has non-negative Kodaira dimension and that the bicanonical map of $S$ factors through the double cover induced by $i.$ It is shown that $S/i$ is regular and either: a) the Albanese fibration of $S$ is of genus 2 or b) $S$ has no genus 2 fibration and $S/i$ is birational to a $K3$ surface. For case a) a list of possibilities and examples are given. An example for case b) with $K^2=6$ is also constructed.