Complete Subvarieties of \rmM_g,n and a Lifting Problem

2023/04/17 by Choi, Daebeom · 1 citation
#14G17 #14H10 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2304.08568

Abstract

Finding the maximal dimension of complete subvarieties of the moduli space of smooth n-pointed curves of genus g is a long-standing open problem. Here we show that for g≥ 3⋅ 2^d-1, if the characteristic of the base field is greater than 2, then \rmM_g contains a complete subvariety of dimension d. Furthermore, in positive characteristic, we construct a complete surface in \rmM_g,n for g≥ 3 and n≥ 1, which contain a general point. These results follow from the proofs of the lifting conjectures, introduced here. In particular, we translate the existence of complete subvarieties to properties of line bundles on \rmM_g,n. Our method reframes Zaal's approach, with increased efficiency via Keel's results on semi-ample line bundles in positive characteristic. This method demonstrates the difference in the geometry of moduli spaces between characteristic 0 and characteristic p.

Cited by

• Extremal effective curves and non-semiample line bundles on \rmM_g,n

Related

• Complete subvarieties in moduli spaces of rank 2 stables sheaves on smooth projective curves and surfaces
• Construction of varieties of low codimension with applications to moduli spaces of varieties of general type
• On the unirationality of moduli spaces of pointed curves
• Density and completeness of subvarieties of moduli spaces of curves or abelian varieties
• Moduli spaces M_g,n(W) for surfaces