Artigo Acesso aberto Revisado por pares

Some unlikely intersections between the Torelli locus and Newton strata in 𝒜 g

2021; Institut de Mathématiques de Bordeaux; Volume: 33; Issue: 1 Linguagem: Inglês

10.5802/jtnb.1159

ISSN

2118-8572

Autores

Joe Kramer-Miller,

Tópico(s)

Polynomial and algebraic computation

Resumo

Let p be an odd prime. What are the possible Newton polygons for a curve in characteristic p? Equivalently, which Newton strata intersect the Torelli locus in 𝒜 g ? In this note, we study the Newton polygons of certain curves with ℤ/pℤ-actions. Many of these curves exhibit unlikely intersections between the Torelli locus and the Newton stratification in 𝒜 g . Here is one example of particular interest: fix a genus g. We show that for any k with 2g 3-2p(p-1) 3≥2k(p-1), there exists a curve of genus g whose Newton polygon has slopes {0,1} g-k(p-1) ⊔{1 2} 2k(p-1) . This provides evidence for Oort’s conjecture that the amalgamation of the Newton polygons of two curves is again the Newton polygon of a curve. We also construct families of curves {C g } g≥1 , where C g is a curve of genus g, whose Newton polygons have interesting asymptotic properties. For example, we construct a family of curves whose Newton polygons are asymptotically bounded below by the graph y=x 2 4g. The proof uses a Newton-over-Hodge result for ℤ/pℤ-covers of curves due to the author, in addition to recent work of Booher–Pries on the realization of this Hodge bound.

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