Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · alg-geom/9607018

Torelli groups and Jacobian varieties of non-orientable compact Klein surfaces

Pablo Ares Gastesi
View original source ↗Read source document ↗Export metadata ↓

Abstract

The Torelli group of a compact non-orientable Klein surface is the subgroup of the modular group consisting of the mapping classes that act trivially on the first homology group of the surface. We prove that if a surface has genus at least $3$, then the Torelli group acts fixed points free on the Teichmüller space of the surface. That gives an embedding of the Torelli space of a Klein surface in the Torelli space of its complex double. We also construct real tori associated to Klein surfaces, which we call the Jacobian of the surface. We prove that this Jacobian is isomorphic to a component of the real part of the Jacobian of the complex double.

Explore related subjects

alg-geommath.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Pablo Ares Gastesi. 1996-07-18. Torelli groups and Jacobian varieties of non-orientable compact Klein surfaces. https://arxiv.org/abs/alg-geom/9607018

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2019

Classification of Varieties with Canonical Curve Section via Gaussian maps on Canonical Curves

Let $C \subset P^{g-1}$ be a smooth canonical curve of genus $g \geq 3$. The purpose of this article is to further develop a method to classify varieties having $C$ as their curve section, using Gaussian map computations. In a previous article a careful analysis of the degeneration to the cone over the hyperplane section was made for _prime_ Fano threefolds, that is Fano threefolds whose Picard group is generated by the hyperplane bundle. In this article we extend this method and classify Fano threefolds of higher index (which still have Picard number one). We are also able to classify Mukai varieties, i.e. varieties of dimension four or more with canonical curve sections.

alg-geom↗
arXiv2017

Boundedness and $K^2$ for log surfaces

Let $ε, C$ be two positive real numbers, and $\mathcal C \subset \mathbb R$ be a DCC (descending chain condition) set. Let $(X, B = \sum b_j B_j)$ denote a projective surface with an $\mathbb R$-divisor. Then (1) The class $\{X\}$ of surfaces for which there exists a divisor $B$ such that $(X,B)$ is $ε$-log terminal and $-(K_X + B)$ is nef (excluding only those for which at the same time $K_X\equiv 0$, $B=0$, and $X$ has at worst Du Val singularities), is bounded. (2) The set $\{(K_X + B)^2\}$ of squares for the semi log canonical pairs $(X, B)$ with ample $K_X + B$ and $b_j \in \mathcal C$, is a DCC set. (3) The class $\{(X,B)\}$ of pairs such that $(X, B)$ is semi log canonical, $K_X + B$ is ample, $(K_X + B)^2 = C$ and $b_j \in \mathcal C$, is bounded.

alg-geom↗