[Paper Review] Seshadri fibrations of algebraic surfaces

Wioletta Syzdek, Tomasz Szemberg|ArXiv.org|Sep 17, 2007

Algebraic Geometry and Number Theory5 references4 citations

TL;DR

This paper refines the relationship between Seshadri constants and fibration structures on algebraic surfaces by showing that the sharp bound from prior work is achieved uniquely by the cubic surface in ℙ³. For all other surfaces, a strictly better bound applies, and the authors establish optimal bounds for multiple-point Seshadri constants that converge to the maximal value as the number of points increases, confirming an asymptotic version of the Nagata-Biran Conjecture.

ABSTRACT

We refine results of Hwang, Keum and Szemberg, Tutaj-Gasinska which relate local invariants - Seshadri constants - of ample line bundles on surfaces to the global geometry - fibration structure. We show that the same picture emerges when looking at Seshadri constants measured at any finite subset of the given surface.

Motivation & Objective

• To refine the known bound on Seshadri constants that force fibration structures on algebraic surfaces.
• To determine whether the sharp bound from previous work applies beyond the cubic surface.
• To extend results on single-point Seshadri constants to multiple-point Seshadri constants on surfaces.
• To establish optimal bounds for multiple-point Seshadri constants and confirm their convergence to the upper bound as the number of points increases.
• To provide asymptotic confirmation of the Nagata-Biran Conjecture for surfaces without fibrations.

Proposed method

• Use of the Hodge Index Theorem to bound the self-intersection of Seshadri curves.
• Application of Lemma 2.3 (Xu) to estimate the self-intersection of curves with prescribed multiplicities at multiple points.
• Employment of Lemma 2.4 (Küchle) to derive numerical inequalities involving multiplicities and the number of points.
• Induction on the number of points to extend results from single-point to multiple-point Seshadri constants.
• Use of the Kodaira-Spencer map to prove rationality of Seshadri curves in the case of multiplicity two.
• Geometric arguments involving blow-ups and nefness of line bundles to characterize Seshadri curves and their fibration properties.

Experimental results

Research questions

• RQ1 Is the cubic surface the only surface for which the sharp bound on Seshadri constants that force fibration is achieved?
• RQ2 Can the bound on Seshadri constants that force fibration be improved for surfaces other than the cubic surface?
• RQ3 How do multiple-point Seshadri constants behave as the number of points increases, and do they converge to the upper bound?
• RQ4 Can the Nagata-Biran Conjecture be asymptotically confirmed for surfaces that do not admit fibrations over curves?
• RQ5 What is the precise structure of Seshadri curves when the Seshadri constant is exactly √((r−1)/r) times the upper bound?

Key findings

• The cubic surface in ℙ³ is the only smooth projective surface for which the Seshadri constant achieves the sharp bound √(3/4)·ε_upper(L;1), and this bound is not improved for any other surface.
• For all surfaces other than the cubic surface, a strictly better bound than √(3/4) applies for single-point Seshadri constants to force fibration.
• The optimal bound for multiple-point Seshadri constants is √((r−1)/r)·ε_upper(L;r), and this bound is sharp for any number of points r.
• For r ≥ 2, the bound √((r−1)/r)·ε_upper(L;r) is optimal and cannot be improved, as shown by examples on ℙ² and rational normal scrolls.
• The Nagata-Biran Conjecture is asymptotically confirmed: for surfaces with no fibration, ε(L;r) ≥ √((r−1)/r)·ε_upper(L;r) for sufficiently large r.
• The bound √((r−1)/r)·ε_upper(L;r) is achieved on ℙ² with r=2 via lines through two points, and on rational normal scrolls via hyperplane sections, showing optimality.