arXiv subjects

Michele Bolognesi

Publications and source records attributed to Michele Bolognesi.

39 records · Page 3Linked to original sources

arXiv2007

Osculating spaces and diophantine equations (with an appendix by Pietro Corvaja and Umberto Zannier)

This paper deals with some classical problems about the projective geometry of complex algebraic curves. We call \textit{locally toric} a projective curve that in a neighbourhood of every point has a local analytical parametrization of type $(t^{a_1},...,t^{a_n})$, with $a_1,..., a_n$ relatively prime positive integers. In this paper we prove that the general tangent line to a locally toric curve in $\bP^3$ meets the curve only at the point of tangency. This result extends and simplifies those of the paper \cite{kaji} by H.Kaji where the same result is proven for any curve in $\bP^3$ such that every branch is smooth. More generally, under mild hypotesis, up to a finite number of anomalous parametrizations $(t^{a_1},...,t^{a_n})$, the general osculating 2-space to a locally toric curve of genus $g<2$ in $\bP^4$ does not meet the curve again. The arithmetic part of the proof of this result relies on the Appendix \cite{cz:rk} to this paper. By means of the same methods we give some applications and we propose possible further developments.

math.AG↗

arXiv2007

A conic bundle degenerating on the Kummer surface

Let $C$ be a genus 2 curve and $\su$ the moduli space of semi-stable rank 2 vector bundles on $C$ with trivial determinant. In \cite{bol:wed} we described the parameter space of non stable extension classes (invariant with respect to the hyperelliptic involution) of the canonical sheaf $ω$ of $C$ with $ω_C^{-1}$. In this paper we study the classifying rational map $ϕ: \pr Ext^1(ω,ω^{-1})\cong \pr^4 \dashrightarrow \su\cong \pr^3$ that sends an extension class on the corresponding rank two vector bundle. Moreover we prove that, if we blow up $\pr^4$ along a certain cubic surface $S$ and $\su$ at the point $p$ corresponding to the bundle $\OO \oplus \OO$, then the induced morphism $\tildeϕ: Bl_S \ra Bl_p\su$ defines a conic bundle that degenerates on the blow up (at $p$) of the Kummer surface naturally contained in $\su$. Furthermore we construct the $\pr^2$-bundle that contains the conic bundle and we discuss the stability and deformations of one of its components.

math.AG↗

arXiv2006

On Weddle Surfaces And Their Moduli

The Weddle surface is classically known to be a birational (partially desingularized) model of the Kummer surface. In this note we go through its relations with moduli spaces of abelian varieties and of rank two vector bundles on a genus 2 curve. First we construct a moduli space A\_2(3)^- parametrizing abelian surfaces with a symmetric theta structure and an odd theta characteristic. Such objects can in fact be seen as Weddle surfaces. We prove that A\_2(3)^- is rational. Then, given a genus 2 curve C, we give an interpretation of the Weddle surface as a moduli space of extensions classes (invariant with respect to the hyperelliptic involution) of the canonical sheaf ωof C with ω^{-1}. This in turn allows to see the Weddle surface as a hyperplane section of the secant variety Sec(C) of the curve C tricanonically embedded in P^4.

math.AG↗