Geometry of ℙ² blown up at seven points

2018/05/10 by Nabanita Ray, Ray, Nabanita
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1805.04064

openalex publication_date 2018/05/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we prove that ℙ² blown up at seven general points admits a conic bundle structure over ℙ¹ and it can be embedded as (2,2) divisor in ℙ¹×ℙ². Conversely, any smooth surface in the complete linear system | (2,2) | of ℙ¹×ℙ² is obtained by blowing up ℙ² at seven points. We also show any smooth surface linearly equivalent to (2,2) in ℙ¹×ℙ² has at most four (-2) curves (the curve which has self intersection (-2)).

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