Extension of the Topological Abel-Jacobi Map for Cubic Threefolds

2023/08/30 by Zhang, Yilong
#14C05 secondary #14J30 #32J05 primary #32S55 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2308.15751

Abstract

The difference [L₁]-[L₂] of a pair of skew lines on a cubic threefold defines a vanishing cycle on the cubic surface as the hyperplane section spanned by the two lines. By deforming the hyperplane, the flat translation of such vanishing cycle forms a 72-to-1 covering space T_v of a Zariski open subspace of (\mathbb P⁴)^*. Based on a lemma of Stein on the compactification of finite analytic covers, we found a compactification of T_v to which the topological Abel-Jacobi map extends. Moreover, the boundary points of the compactification can be interpreted in terms of local monodromy and the singularities on cubic surfaces. We prove the associated map on fundamental groups of topological Abel-Jacobi map is surjective.

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