EMS Magazine Article from: Anita Waltho

### Abstract D. Mumford conjectured in [33] that the rational cohomology of the stable moduli space of Riemann surfaces is a polynomial algebra generated by certain classes $\kappa_i$ of dimension $2i$. For the purpose of calculating rational cohomology, one may replace the stable moduli space of Riemann surfaces by $B\Gamma_{\infty}$, where $\Gamma_\infty$ is the group of isotopy classes of automorphisms of a smooth oriented connected surface of “large” genus. Tillmann’s theorem [44] that the plus construction makes $B\Gamma_{\infty}$ into an infinite loop space led to a stable homotopy version of Mumford’s conjecture, stronger than the original [24]. We prove the stronger version, relying on Harer’s stability theorem [17], Vassiliev’s theorem concerning spaces of functions with moderate singularities [46], [45] and methods from homotopy theory.

Classification of combinatorial objects. Contribute to abetten/orbiter development by creating an account on GitHub.

We determine the number of cubic surfaces with 27 lines over a finite field $${{\mathbb {F}}}_q$$ F q . This is based on exploiting the relationship between non-conical six-arcs in a projective plane embedded in projective three-space and cubic surfaces with 27 lines. We revisit this classical relationship, which goes back to work of Clebsch in the nineteenth century. Our result can be used as an enumerative check for a computer classification of cubic surfaces with 27 lines over finite fields.