Let $N\ge 1$ be an integer. Let $X_0\left(N\right)$ be the modular curve over $\mathbf{Z}$, as constructed by Katz and Mazur. The minimal resolution of $X_0\left(N\right)$ over $\mathbf{Z}[1/6]$ is computed. Let $p\ge 5$ be a prime, such that $N=p^{2}M$, with $M$ prime to $p$. Let $n=(p^2-1)/2$. It is shown that $X_0\left(N\right)$ has stable reduction at $p$ over $\mathbf{Q}\left[\sqrt[n]{p}\right]$, and the fibre at $p$ of the stable model is computed.

We study general elements of moduli spaces ${\mathfrak{M}}_{{\mathbb{P}}^2}\left(2,c_1,c_2\right)$ of rank-2 stable holomorphic vector bundles on ${\mathbb{P}}^2$ and their minimal free resolutions. Incidentally, a quite easy proof of the irreducibility of ${\mathfrak{M}}_{{\mathbb{P}}^2}\left(2,c_1,c_2\right)$ is shown.

A combinatorial description of the minimal free resolution of a lattice ideal allows us to the connection of Integer Linear Programming and Al1gebra. The non null reduced homology spaces of some simplicial complexes are the key. The extremal rays of the associated cone reduce the number of variables.

Sia $p:X\to P^2$ un fibrato in coniche standard con curva discriminante $\mathrm{\Delta }$ di grado $d$. La varietà delle sezioni minime delle superfici $p^{-1}\left(C\right)$, dove $C$ è una curva di grado $d-3$, si spezza in due componenti ${\mathcal{C}}_{+}$ e ${\mathcal{C}}_{-}$. Si prova che, mediante la mappa di Abel-Jacobi $\mathrm{\Phi }$, una di queste componenti domina la Jacobiana intermedia $JX$, mentre l'altra domina il divisore theta $\mathrm{\Theta }\subset JX$. Questi risultati vengono applicati ad alcuni threefold di Fano birazionalmente equivalenti a un fibrato in coniche. In particolare si prova che il generico...

Soient $ℰ$ un fibré de rang 2 sur l’espace projectif de dimension 3 sur un corps algébriquement clos et $n$ un entier tel que $H^0ℰ(n-1)=0$ et $H^0ℰ\left(n\right)\ne 0$. Toute courbe $C$ schéma des zéros d’une section non nulle de $ℰ\left(n\right)$ est une courbe minimale dans sa classe de biliaison.

1. Introduction. Dans un article célèbre, D. H. Lehmer posait la question suivante (voir [Le], §13, page 476): «The following problem arises immediately. If ε is a positive quantity, to find a polynomial of the form: $f\left(x\right)=x^r+a_1{x}^{r-1}+\cdots +a_r$ where the a’s are integers, such that the absolute value of the product of those roots of f which lie outside the unit circle, lies between 1 and 1 + ε (...). Whether or not the problem has a solution for ε < 0.176 we do not know.» Cette question, toujours ouverte, est la source...