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We compute the rational cohomology ring of \bar R_2, the (compactified)moduli space of Prym curves of genus 2. We also recompute the rationalcohomology ring of \bar S_2, the moduli space of spin curves of genus 2,thereby correcting some errors made in an article by G. Bini and C. Fontanari,containing a computation of the same ring.

Rational cohomology of \bar R_2 (and \bar S_2)