Tautological classes on the moduli space of hyperelliptic curves with rational tails

We study tautological classes on the moduli space of stable n-pointed hyperelliptic curves of genus g with rational tails. The method is based on the approach of Yin in comparing tautological classes on the moduli of curves and the universal Jacobian. Our result gives a complete description of tautological relations. It is proven that all relations come from the Jacobian side. The intersection pairings are shown to be perfect in all degrees. We show that the tautological algebra coincides with its image in cohomology via the cycle class map. The latter is identified with monodromy invariant classes in cohomology.