### Working paper

## Computation of the first Stiefel-Whitney class for the variety $\overline{{\mathcal M}_{0,n}^{\mathbb R}}$

Using meromorphic differentials with real periods, we prove Arbarello's conjecture that any compact complex cycle of dimension g−n in the moduli space M_g of smooth algebraic curves of genus g must intersect the locus of curves having a Weierstrass point of order at most n.

We show that a certain moduli space of minimal A∞-structures coincides with the modular compactification ℳ_{1,n(n−1)} of ℳ_{1,n} constructed by Smyth in [26]. In addition, we describe these moduli spaces and the universal curves over them by explicit equations, prove that they are normal and Gorenstein, show that their Picard groups have no torsion and that they have rational singularities if and only if n≤11.

In this paper we study the moduli stack M_{1,n} of curves of arithmetic genus 1 with n marked points, forming a nonspecial divisor. In Polishchuk (A modular compactification of M_{1,n} from A∞-structures, arXiv:1408.0611) this stack was realized as the quotient of an explicit scheme U^{ns}_{1,n} affine of finite type over ℙn−1, by the action of 𝔾m^n . Our main result is an explicit description of the corresponding GIT semistable loci in U^{ns}_{1,n}. This allows us to identify some of the GIT quotients with some of the modular compactifications of M_{1,n} defined in Smyth (Invent Math 192:459–503, 2013; Compos Math 147(3):877–913, 2011).

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.