Cuspidal quintics and surfaces with pg=0,p_g=0, K2=3K^2=3 and 5-torsion

If $S$ is a quintic surface in $\mathbb P^3$ with singular set $15$ $3$-divisible ordinary cusps, then there is a Galois triple cover $\phi:X\to S$ branched only at the cusps such that $p_g(X)=4,$ $q(X)=0,$ $K_X^2=15$ and $\phi$ is the canonical map of $X$. We use computer algebra to search for such quintics having a free action of $\mathbb Z_5$, so that $X/{\mathbb Z_5}$ is a smooth minimal surface of general type with $p_g=0$ and $K^2=3$. We find two different quintics, one of which is the Van der Geer--Zagier quintic, the other is new. We also construct a quintic threefold passing through the $15$ singular lines of the Igusa quartic, with $15$ cuspidal lines there. By taking tangent hyperplane sections, we compute quintic surfaces with singular set $17\mathsf A_2$, $16\mathsf A_2$, $15\mathsf A_2+\mathsf A_3$ and $15\mathsf A_2+\mathsf D_4$.Comment: Exposition improved according to the Referee suggestions. Final versio

On surfaces with pg=q=1p_g=q=1 and non-ruled bicanonical involution

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.Comment: revised version, correction in main theorem, to appear in Ann. Scuola Norm. Sup. Pis

Tangible storytelling: let children play with the bits

The use of tangible objects makes it possible to create interactions, or dynamics, which are alternatives to the mouse and keyboard in the process of communicating with the computer. The construction of these objects incorporating electronic components lets us bring that momentum to another level. This meeting with the technology allows children to take an active role, while there is a purpose of control over the objects, which becomes important to them. With the reinforcement of that control, the introduction of programmable digital electronic components also allows the child to develop, strengthen and feel the impact of their role as competent designer and creator of technology. Current technology allows the construction of these objects and the communication with computers at a low cost through micro-controllers, using, on one hand, the open source software and on the other the open hardware.info:eu-repo/semantics/publishedVersio

A note on Todorov surfaces

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}.Comment: 9 page

A surface with canonical map of degree 2424

We construct a complex algebraic surface with geometric genus $p_g=3$, irregularity $q=0$, self-intersection of the canonical divisor $K^2=24$ and canonical map of degree $24$ onto $\mathbb P^2$.Comment: Minor changes, according to the Referee comments. Final versio