In this paper, we give an explicit presentation of the integral Chow ring of a stack of smooth plane cubics. We also determine some relations in the general case of hypersurfaces of any dimension and degree.

This paper is about the cohomology of certain finite-index subgroups of mapping class groups and its relation to the cohomology of arithmetic groups. For $G=\mathbb{Z}/m\mathbb{Z}$ and for a regular $G$-cover $S\to \overline{S}$ (possibly branched), a finite-index subgroup $\Gamma <Mod\left(\overline{S}\right)$ acts on $H_1(S;\mathbb{Z})$ commuting with the deck group action, thus inducing a homomorphism $\Gamma \to {Sp}_{2g}^G\left(\mathbb{Z}\right)$ to an arithmetic group. The induced map $H^{\ast }\left({Sp}_{2g}^G\right(\mathbb{Z});\mathbb{Q})\to H^{\ast }(\Gamma ;\mathbb{Q})$ can be understood using index theory. To this end, we describe a families version of the $G$-index theorem for the signature operator and apply this to (i) compute $H^2\left({Sp}_{2g}^G\right(\mathbb{Z});\mathbb{Q})\to H^2(\Gamma ;\mathbb{Q})$, (ii) rederive Hirzebruch’s formula for signature of a branched cover, (iii) compute Toledo invariants of surface group representations to $SU(p,q)$ arising from Atiyah–Kodaira constructions, and (iv) describe how classes in $H^{\ast }\left({Sp}_{2g}^G\right(\mathbb{Z});\mathbb{Q})$ give equivariant cobordism invariants for surface bundles with a fiberwise $G$ action, following Church–Farb–Thibault.

By considering negative surgeries on a knot $K$ in $S^3$, we derive a lower bound on the nonorientable slice genus ${\gamma }_4\left(K\right)$ in terms of the signature $\sigma \left(K\right)$ and the concordance invariants $V_i\left(\overline{K}\right)$; this bound strengthens a previous bound given by Batson and coincides with Ozsváth–Stipsicz–Szabó’s bound in terms of their $\upsilon$ invariant for L-space knots and quasi-alternating knots. A curious feature of our bound is superadditivity, implying, for instance, that the bound on the stable nonorientable slice genus is sometimes better than that on ${\gamma }_4\left(K\right)$.

We classify all complex surfaces with quotient singularities that do not contain any smooth rational curves under the assumption that the canonical divisor of the surface is not pseudo-effective. As a corollary, we show that if $X$ is a log del Pezzo surface such that, for every closed point $p\in X$, there is a smooth curve (locally analytically) passing through $p$, then $X$ contains at least one smooth rational curve.

Using invariant Zariski–Riemann spaces, we prove that every normal toric variety over a valuation ring of rank one can be embedded as an open dense subset into a proper toric variety equivariantly. This extends a well-known theorem of Sumihiro for toric varieties over a field to this more general setting.

We generalize the Bernstein–Sato polynomials of Budur, Mustaţǎ, and Saito to ideals in normal semigroup rings. In the case of monomial ideals, we also relate the roots of the Bernstein–Sato polynomial to the jumping coefficients of the corresponding multiplier ideals. To prove the latter result, we obtain a new combinatorial description for the multiplier ideals of a monomial ideal in a normal semigroup ring.

The BNSR-invariants of a group $G$ are a sequence ${\Sigma }^1\left(G\right)\supseteq {\Sigma }^2\left(G\right)\supseteq \cdots$ of geometric invariants that reveal important information about finiteness properties of certain subgroups of $G$. We consider the symmetric automorphism group ${\Sigma Aut}_n$ and pure symmetric automorphism group ${P\Sigma Aut}_n$ of the free group $F_n$ and inspect their BNSR-invariants. We prove that for $n\ge 2$, all the “positive” and “negative” character classes of ${P\Sigma Aut}_n$ lie in ${\Sigma }^{n-2}\left({P\Sigma Aut}_n\right)\setminus {\Sigma }^{n-1}\left({P\Sigma Aut}_n\right)$. We use this to prove that for $n\ge 2$, ${\Sigma }^{n-2}\left({\Sigma Aut}_n\right)$ equals the full character sphere $S^0$ of ${\Sigma Aut}_n$ but ${\Sigma }^{n-1}\left({\Sigma Aut}_n\right)$ is empty, so in particular the commutator subgroup ${\Sigma Aut}_n\text{'}$ is of type $F_{n-2}$ but not $F_{n-1}$. Our techniques involve applying Morse theory to the complex of symmetric marked cactus graphs.

Let $(A,\mathfrak{m})$ be a Cohen–Macaulay local ring, and let $I$ be an ideal of $A$. We prove that the Rees algebra $\mathcal{R}\left(I\right)$ is an almost Gorenstein ring in the following cases:

(1) $(A,\mathfrak{m})$ is a two-dimensional excellent Gorenstein normal domain over an algebraically closed field $K\cong A/\mathfrak{m}$, and $I$ is a $p_g$-ideal;

(2) $(A,\mathfrak{m})$ is a two-dimensional almost Gorenstein local ring having minimal multiplicity, and $I={\mathfrak{m}}^{\ell }$ for all $\ell \ge 1$;

(3) $(A,\mathfrak{m})$ is a regular local ring of dimension $d\ge 2$, and $I={\mathfrak{m}}^{d-1}$. Conversely, if $\mathcal{R}\left({\mathfrak{m}}^{\ell }\right)$ is an almost Gorenstein graded ring for some $\ell \ge 2$ and $d\ge 3$, then $\ell =d-1$.

In an influential 2008 paper, Baker proposed a number of conjectures relating the Brill–Noether theory of algebraic curves with a divisor theory on finite graphs. In this note, we examine Baker’s Brill–Noether existence conjecture for special divisors. For $g\le 5$ and $\rho (g,r,d)$ nonnegative, every graph of genus $g$ is shown to admit a divisor of rank $r$ and degree at most $d$. As further evidence, the conjecture is shown to hold in rank $1$ for a number families of highly connected combinatorial types of graphs. In the relevant genera, our arguments give the first combinatorial proof of the Brill–Noether existence theorem for metric graphs, giving a partial answer to a related question of Baker.

We generalize the Karrass–Pietrowski–Solitar and the Nielsen realization theorems from the setting of free groups to that of free products. As a result, we obtain a fixed point theorem for finite groups of outer automorphisms acting on the relative free splitting complex of Handel and Mosher and on the outer space of a free product of Guirardel and Levitt, and also a relative version of the Nielsen realization theorem, which, in the case of free groups, answers a question of Karen Vogtmann. We also prove Nielsen realization for limit groups and, as a byproduct, obtain a new proof that limit groups are CAT($0$).

The proofs rely on a new version of Stallings’ theorem on groups with at least two ends, in which some control over the behavior of virtual free factors is gained.