6533b827fe1ef96bd1286f5f

RESEARCH PRODUCT

An Arakelov inequality in characteristic p and upper bound of p-rank zero locus

Mao ShengJun LuKang Zuo

subject

Abelian varietyAlgebra and Number TheoryStable curveCombinatoricsAlgebraic cycleMathematics - Algebraic GeometryMathematics::Algebraic Geometry14D05 (Primary) 14G25 14H10 (Secondary)Algebraic surfaceFOS: MathematicsGenus fieldAlgebraic curveAbelian groupAlgebraic Geometry (math.AG)Singular point of an algebraic varietyMathematics

description

In this paper we show an Arakelov inequality for semi-stable families of algebraic curves of genus $g\geq 1$ over characteristic $p$ with nontrivial Kodaira-Spencer maps. We apply this inequality to obtain an upper bound of the number of algebraic curves of $p-$rank zero in a semi-stable family over characteristic $p$ with nontrivial Kodaira-Spencer map in terms of the genus of a general closed fiber, the genus of the base curve and the number of singular fibres. An extension of the above results to smooth families of Abelian varieties over $k$ with $W_2$-lifting assumption is also included.

yearjournalcountryeditionlanguage
2008-11-10Journal of Number Theory
https://doi.org/10.1016/j.jnt.2009.05.015