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Quadratic families of elliptic curves and unirationality of degree 1 conic bundles
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 139, Number 4, August 2017
- pp. 915-936
- 10.1353/ajm.2017.0024
- Article
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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.
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