THE STRONG FRANCHETTA CONJECTURE IN ARBITRARY CHARACTERISTICS

STEFAN SCHRÖER
2003 International Journal of Mathematics  
Introduction Let M g be the coarse moduli space of smooth curves of genus g ≥ 3 over an arbitrary ground field k. Deligne and Mumford [9] showed that the normal quasiprojective scheme M g is irreducible. Let η ∈ M g be its generic point and C = M g,1 → M g the tautological curve. The generic fiber C η is a smooth curve of genus g over the function field κ(η) of the moduli space M g . We call it the generic curve. Franchetta [12] conjectured Pic(C η ) = ZK Cη . Arbarello and Cornalba [2] proved
more » ... ornalba [2] proved this over the complex numbers using Harer's calculation [28] of the second homology for the mapping class group of Riemann surfaces. The latter is a purely topological result. Later, Arbarello and Cornalba [3] gave an algebro-geometric proof over the complex numbers. Mestrano [40] and Kouvidakis [36] deduced the Strong Franchetta Conjecture over C, which states that the rational points in the Picard scheme Pic Cη/η are precisely the multiples of the canonical class. The first goal of this paper is to give an algebraic proof for the Strong Franchetta Conjecture in all characteristics p ≥ 0. The idea is to construct special stable curves showing that any divisor class violating Franchetta's Conjecture must be nontorsion. Having this, we use Moriwaki's recent calculation [42] of Pic(M g,n+1 ) ⊗ Q in characteristic p > 0 to infer the Strong Franchetta Conjecture. I also prove the Franchetta Conjecture for generic pointed curves: Their Picard groups are freely generated by the canonical class and the marked points. Actually, I use the pointed case as an essential step in the proof for the unpointed case. The second goal of this paper is to show that there are many other nonclosed points x ∈ M g,n such that the marked points and the canonical class generates Pic(C x ), at least up to torsion. This seems to be new even in characteristic zero. We shall see that over uncountable ground fields, there is an uncountable dense set of such points with dim {x} ≤ 2. This relies on Hilbert's Irreducibility Theorem for function fields. The idea is to view C η as the generic fiber of some fibered surface Y , extend this to a family of fibered surfaces Y → S over some parameter space S, and apply Hilbert's Irreducibility Theorem and the Tate-Shioda Formula to the resulting family of Néron-Severi scheme s → NS Ys/s . Such specialization arguments are problematic in characteristic p > 0, because ungeometric properties such as regularity behave badly in families. However, we overcome these difficulties by using the theory of geometric unibranch singularities.
doi:10.1142/s0129167x03001752 fatcat:rjw3smi6nfggnmpgrqfiupmtte