Let ${\mathcal {R}} \subset \mathbb {P}^1_{\mathbb {C}}$ be a finite subset of markings. Let G be an almost simple simply-connected algebraic group over $\mathbb {C}$. Let $K_G$ denote the compact real form of G. Suppose for each lasso l around the marked point, a conjugacy class $C_l$ in $K_G$ is prescribed. The aim of this paper is to give verifiable criteria for the existence of an irreducible homomorphism of $\pi _{1}(\mathbb P^1_{\mathbb {C}} \,{\backslash}\, {\mathcal {R}})$ into $K_G$ such that the image of l lies in $C_l$.

Using the theory of cohomological invariants for algebraic stacks, we compute the Brauer group of the moduli stack of hyperelliptic curves ${\mathcal {H}}_g$ over any field of characteristic $0$. In positive characteristic, we compute the part of the Brauer group whose order is prime to the characteristic of the base field.

We prove a decomposition formula of logarithmic Gromov–Witten invariants in a degeneration setting. A one-parameter log smooth family $X \longrightarrow B$ with singular fibre over $b_0\in B$ yields a family $\mathscr {M}(X/B,\beta ) \longrightarrow B$ of moduli stacks of stable logarithmic maps. We give a virtual decomposition of the fibre of this family over $b_0$ in terms of rigid tropical maps to the tropicalization of $X/B$. This generalizes one aspect of known results in the case that the fibre $X_{b_0}$ is a normal crossings union of two divisors. We exhibit our formulas in explicit examples.

We find upper bounds on the essential dimension of the moduli stack of parabolic vector bundles over a curve. When there is no parabolic structure, we improve the known upper bound on the essential dimension of the usual moduli stack. Our calculations also give lower bounds on the essential dimension of the semistable locus inside the moduli stack of vector bundles of rank r and degree d without parabolic structure.

The theory of generalized elliptic curves gives a moduli-theoretic compactification for modular curves when the level is a unit on the base, and the theory of Drinfeld structures on elliptic curves provides moduli schemes over the integers without a modular interpretation of the cusps. To unify these viewpoints it is natural to consider Drinfeld structures on generalized elliptic curves, but some of these resulting moduli problems have non-étale automorphism groups and so cannot be Deligne–Mumford stacks. Artin’s method as used in the work of Deligne and Rapoport rests on a technique of passage to irreducible fibers (where the geometry determines the group theory), and this does not work in the presence of non-étale level structures and non-étale automorphism groups. By making more efficient use of the group theory to bypass these difficulties, we prove that the standard moduli problems for Drinfeld structures on generalized elliptic curves are proper Artin stacks. We also analyze the local structure on these stacks and give some applications to Hecke correspondences.