arXiv subjects

Jean Vallès

Publications and source records attributed to Jean Vallès.

25 records · Page 2Linked to original sources

arXiv2012

A vector bundle proof of Poncelet theorem

In the town of Saratov where he was prisonner, Poncelet, continuing the work of Euler and Steiner on polygons simultaneously inscribed in a circle and circumscribed around an other circle, proved the following generalization : "Let C and D be two smooth conics in the projective complex plane. If D passes through the n(n-1)/2 vertices of a complete polygon with n sides tangent to C then D passes through the vertices of infinitely many such polygons." According to Marcel Berger this theorem is the nicest result about the geometry of conics. Even if it is, there are few proofs of it. To my knowledge there are only three. The first proof, published in 1822 and based on infinitesimal deformations, is due to Poncelet. Later, Jacobi proposed a new proof based on finite order points on elliptic curves; his proof, certainly the most famous, is explained in a modern way and in detail by Griffiths and Harris. In 1870 Weyr proved a Poncelet theorem in space (more precisely for two quadrics) that implies the one above when one quadric is a cone; this proof is explained by Barth and Bauer. Our aim in this short note is to involve vector bundles techniques to propose a new proof of this celebrated result. Poncelet did not appreciate Jacobi's for the reason that it was too far from the geometric intuition. I guess that he would not appreciate our proof either for the same reason.

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arXiv2010

Hyperplane arrangements of Torelli type

We give a necessary and sufficient condition in order for a hyperplane arrangement to be of Torelli type, namely that it is recovered as the set of unstable hyperplanes of its Dolgachev sheaf of logarithmic differentials. Decompositions and semistability of non-Torelli arrangements are investigated.

math.AG↗

arXiv2010

Fibrés de Schwarzenberger et fibrés logarithmiques généralisés

We propose a generalization of logarithmic and Schwarzenberger bundles over $¶^n=¶^n(\C)$ when the rank is greater than $n$. The first ones are associated to finite sets of points on $¶^{n\vee}$ and the second ones to curves with degree greater than $n$ on $¶^{n\vee}$. On the projective plane we show that two logarithmic bundles are isomorphic if and only if they are associated to the same set of points or if the two sets of points belong to a curve of degree equal to the rank of the considered bundles.

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arXiv2008

Fibrés vectoriels de rang deux sur $¶^2$ provenant d'un revêtement double

Since Schwarzenberger and his celebrated paper called "Vector bundles on the projective plane" we know that any rank two vector bundle on $¶^2$ is a direct image of a line bundle on a double covering of the plane. This theorem suggests to study the rank two vector bundles according to the branch curve of the covering which they come from. Thus, in the first part we prove that, given a double covering ramified over an irreducible curve $C_{2r}$ with degree $2r$, the jumping lines of fixed order (order depending on $r$ and on the parity of the rank two vector bundle) of the direct images vector bundles are necessarely $r$-tangent to $C_{2r}$. In the second part we concentrate on the case $r=2$. Then we give a list of vector bundles for which the jumping lines are exactly the bitangent lines to the branch quartic.

math.AG↗

arXiv2008

Geometry of syzygies via Poncelet varieties

We consider the Grassmannian $\mathbb{G}r(k,n)$ of $(k+1)$-dimensional linear subspaces of $V_n=H^0({¶^1},Ø_{¶^1}(n))$. We define $\frak{X}_{k,r,d}$ as the classifying space of the $k$-dimensional linear systems of degree $n$ on $¶^1$ whose basis realize a fixed number of polynomial relations of fixed degree, say a fixed number of syzygies of a certain degree. The first result of this paper is the computation of the dimension of $\frak{X}_{k,r,d}$. In the second part we make a link between $\frak{X}_{k,r,d}$ and the Poncelet varieties. In particular, we prove that the existence of linear syzygies implies the existence of singularities on the Poncelet varieties.

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arXiv2006

Variétés de type Togliatti

The Del Pezzo surface S\_6 obtained by blowing up the projective plane along three points (non aligned) has the following nice property : its osculating hyperplanes have a common point. For that reason S\_6 is also called Togliatti surface. In this paper we give two explanations of this fact. After what we obtain two different families of varieties with an analogous property.

math.AG↗

arXiv2005

Fibrés logarithmiques sur le plan projectif

We describe the scheme of jumping lines of logarithmic vector bundles on the projective plane. This result is already proved by Dolgachev and Kapranov when the first Chern class is even, it is new when the first Chern class is odd.

math.AG↗