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

Quadratic families of elliptic curves and unirationality of degree 1 conic bundles - János Kollár

János Kollár
Princeton University
April 13, 2015
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. (joint work with M. Mella)
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