×

A generalisation of Nagata’s theorem on ruled surfaces. (English) Zbl 1047.14018

From the introduction: Let \(X\) be a smooth projective irreducible curve of genus \(g\) over an algebraically closed field \(k\), \(G\) a connected reductive algebraic group over \(k\) and \(P\) a parabolic subgroup. For a principal \(G\)-bundle \(E\) over \(X\), consider the associated \(G/P\)-bundle \(\pi: E/P\to X\). If \(\sigma\) is a section of \(\pi\) we denote by \(N_\sigma\) the normal bundle of \(\sigma(X)\) in \(E/P\). The first result proved in this paper is the following:
Theorem 1.1. There exists a section \(\sigma\) of \(\pi:E/P\to X\) such that \(\deg(N_\sigma)\leq g\cdot\dim(G/P)\), where \(g\) is the genus of \(X\), and \(\deg (N_\sigma)\) denotes the degree of the normal bundle \(N_\sigma\) considered as a vector bundle on \(X\).
The above result was classically known in the case of \(G=GL(2)\) and \(P\) a maximal parabolic, in the form of the theorem of M. Nagata [Nagoya Math. J. 37, 191–196 (1970; Zbl 0193.21603)] and C. Segre, which asserts that a ruled surface on \(X\) admits a section whose self intersection number is \(\leq g\). It has also been proved for \(G=GL(n)\) and \(P\) a maximal parabolic subgroup by Mukai and Sakai, and for \(G\) a classical group and \(P\) a maximal parabolic subgroup by Nitsure.
In the second part of the paper, we prove the following:
Theorem 1.2. Let \(G\) be a connected reductive algebraic group and \(X\) a smooth projective irreducible curve over an algebraically closed field \(k\) of arbitrary characteristic. Then the set of isomorphism classes of semi-stable \(G\)-bundles on the curve \(X\) with a given degree is bounded. In particular, if \(G\) is semi-simple then semi-stable \(G\)-bundles form a bounded family.
In characteristic 0, the above theorem is due to Ramanathan. For the classical groups, the result follows in all characteristics (except in characteristic 2 for \(G=\text{SO}(n))\) from the observation of Ramanan that a \(G\)-bundle is semi-stable if and only if the underlying vector bundle is so.

MSC:

14H60 Vector bundles on curves and their moduli
14H45 Special algebraic curves and curves of low genus
14L30 Group actions on varieties or schemes (quotients)
14M17 Homogeneous spaces and generalizations

Citations:

Zbl 0193.21603
PDFBibTeX XMLCite
Full Text: DOI