Quadratic families of elliptic curves and unirationality of degree 1   conic bundles
  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. 
J\'anos Koll\'ar (Princeton Univ.) and Massimiliano Mella (Univ. di   Ferrara)
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