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

János Kollár, Massimiliano Mella

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Abstract

We study families of elliptic curves whose coefficients are degree 2 polynomials in a variable t. All such curves together form an algebraic surface which is birational to a conic bundle with 7 singular fibers. We prove that such conic bundles are unirational. As a consequence one obtains that, for infinitely many values of t, the resulting elliptic curve has rank at least 1.

Original languageEnglish (US)
Pages (from-to)915-936
Number of pages22
JournalAmerican Journal of Mathematics
Volume139
Issue number4
DOIs
StatePublished - 2017

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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