Quadratic families of elliptic curves and unirationality of degree 1 conic bundles

2014/12/11 by János Kollár, Massimiliano Mella, Kollár, János +1 · 4 citations
Mathematics · #11G05 (Primary) #11G35 #14E08 (Secondary) #14J26 #14M20 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT #msc:11G05 #msc:11G35 #msc:14E08 #msc:14J26 #msc:14M20

paper · pdf · doi:10.48550/arxiv.1412.3673

arxiv created 2016/04/11 · arxiv updated 2016/04/12

Abstract

We consider elliptic curves whose coefficients are degree 2 polynomials in a variable t. We prove that for infinitely many values of t the resulting elliptic curve has rank at least 1. All such curves together form an algebraic surface which is birational to a conic bundle with 7 singular fibers. The main step of the proof is to show that such conic bundles are unirational. V.2: Main theorem is corrected for small finite fields.

Cited by

• Unboundedness of zero-cycles on higher dimensional Fano manifolds
• On the unirationality of conic bundles with discriminant of degree eight
• Rank 2 vector bundles and degrees of points of del Pezzo surfaces
• Degree two unirational parametrizations over the real field

Related

• Conic bundles and Mordell--Weil ranks of elliptic surfaces
• On the unirationality of conic bundles with discriminant of degree eight
• An elliptic conic bundle in P⁴ arising from a stable rank-3 vector bundle
• Elliptic curves in moduli space of stable bundles
• Two infinite families of elliptic curves with rank greater than one