Algebraic surfaces with p_g=q=1, K²=4 and nonhyperelliptic Albanese fibrations of genus 4

2017/11/29 by Songbo Ling, Ling, Songbo
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #math.AG

paper · pdf · doi:10.48550/arxiv.1711.10650

16 pages

arxiv created 2017/11/29 · openalex publication_date 2017/11/29 · arxiv updated 2017/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we study minimal algebraic surfaces with p_g=q=1,K²=4 and nonhyperelliptic Albanese fibrations of genus 4. We construct for the first time a family of such surfaces as complete intersections of type (2,3) in a ℙ³-bundle over an elliptic curve. For the surfaces we construct here, the direct image of the canonical sheaf under the Albanese map is decomposable (which is a topological invariant property). Moreover we prove that, all minimal surfaces with p_g=q=1,K²=4 and nonhyperelliptic Albanese fibrations of genus 4 such that the direct image of the canonical sheaf under the Albanese map is decomposable are contained in our family. As a consequence, we show that these surfaces constitute a 4-dimensional irreducible subset M of M_1,1^4,4, the Gieseker moduli space of minimal surfaces with p_g=q=1, K²=g=4. Moreover, the closure of M is an irreducible component of M_1,1^4,4.

Citations

• Vector Bundles Over an Elliptic Curve

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