The Szpiro inequality for higher genus fibrations
3 pages
English

The Szpiro inequality for higher genus fibrations

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3 pages
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The Szpiro inequality for higher genus fibrations Arnaud BEAUVILLE Introduction The aim of this note is to prove the following result: Proposition .? Let f : S ? B be a non-trivial semi-stable fibration of genus g ≥ 2 , N the number of critical points of f and s the number of singular fibres. Then N < (4g + 2)(s + 2g(B)? 2) . Recall that a semi-stable fibration of genus g is a surjective holomorphic map of a smooth projective surface S onto a smooth curve B , whose generic fibre is a smooth curve of genus g and whose singular fibres are allowed only ordinary double points; moreover we impose that each smooth rational curve contained in a fibre meets the rest of the fibre in at least 2 points (otherwise by blowing up non-critical points of f in a singular fibre we could arbitrarily increase N keeping s fixed). The corresponding inequality N ≤ 6(s + 2g(B)? 2) in the case g = 1 has been observed by Szpiro; it was motivated by the case of curves over a number field, where an analogous inequality would have far-reaching consequences [S]. The higher genus case is considered in the recent preprint [BKP], where the authors prove the slightly weaker inequality N ≤ (4g + 2)s for hyperelliptic fibrations over P1 .

  • class inequality

  • semi-stable fibration

  • ?f

  • fibered algebraic

  • trivial semi

  • hyperelliptic fibrations

  • fibration over


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The Szpiro inequality for higher genus fibrations Arnaud BEAUVILLE
Introduction The aim of this note is to prove the following result: Proposition.Letf: SBbe a non-trivial semi-stable fibration of genusg2, Nthe number of critical points offandsthe number of singular fibres.Then N<(4g+ 2)(s+ 2g(B)2). Recall that a semi-stable fibration of genusgis a surjective holomorphic map of a smooth projective surfaceS ontoa smooth curveB ,whose generic fibre is a smooth curve of genusgand whose singular fibres are allowed only ordinary double points; moreover we impose that each smooth rational curve contained in a fibre meets the rest of the fibre in at least 2 points (otherwise by blowing up non-critical points offin a singular fibre we could arbitrarily increaseN keepingsfixed). The corresponding inequalityN6(s+ 2g(B)the case2) inghas been= 1 observed by Szpiro; it was motivated by the case of curves over a number field, where an analogous inequality would have far-reaching consequences [S]. The higher genus case is considered in the recent preprint [BKP], where the authors prove the slightly 1 weaker inequalityN(4g+ 2)sfor hyperelliptic fibrations overPmethod. Their is topological, and in fact the result applies in the much wider context of symplectic Lefschetz fibrations.We will show that in the more restricted algebraic-geometric set-up, the Proposition is a direct consequence of two classical inequalities in surface theory. Itwould be interesting to know whether the proof of [BKP] can be extended to non-hyperelliptic fibrations.
Proof The main numerical invariants of a surfaceS arethe square of the canonical bun-dle KScce´rarahretcitsiniac-roPuEelt,ehχ(OSan)hedtpotoncar´eoligacEllureP-io 2 characteristice(S) ;they are linked by the Noether formula12χ(OS) = K+e(S) . S For a semi-stable fibrationf: SB ithas become customary to modify these in-∗ −1 variants as follows.Letbbe the genus ofand KB ,f= KXfK therelative B canonical bundle ofX over B; then we consider: 2 2 K =K8(b1)(g1) fX χf:= degf(Kf) =χ(OX)(b1)(g1) ef:= N =e(X)4(b1)(g1). 1
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