When the genus g is even, we extend the computation of mod 2 cohomological
invariants of Hg to non algebraically closed fields, we give an
explicit functorial description of the invariants and we completely describe
their multiplicative structure. In the Appendix, we show that the cohomological
invariants of the compactification Hg are trivial, and
use our methods to give a very short proof of a result by Cornalba on the
Picard group of the compactification Hg and extend it
to positive characteristicComment: 25 pages, comments are welcome! To appear on Documenta Mathematic