We prove that limits of multiplicities associated to graded families of ideals exist under very general conditions. Most of our results hold for analytically unramified equicharacteristic local rings with perfect residue fields. We give a number of applications, including a “$\text{ volume}=\text{ multiplicity}$” formula, generalizing the formula of Lazarsfeld and Mustaţă, and a proof that the epsilon multiplicity of Ulrich and Validashti exists as a limit for ideals in rather general rings, including analytic local domains. We prove a generalization of this to generalized symbolic powers of ideals proposed by Herzog, Puthenpurakal and Verma. We also prove an asymptotic “additivity formula” for limits of multiplicities and a formula on limiting growth of valuations, which answers a question posed by the author, Kia Dalili and Olga Kashcheyeva. Our proofs are inspired by a philosophy of Okounkov for computing limits of multiplicities as the volume of a slice of an appropriate cone generated by a semigroup determined by an appropriate filtration on a family of algebraic objects.

For a noncyclic finite group $G$, let $\gamma \left(G\right)$ denote the smallest number of conjugacy classes of proper subgroups of $G$ needed to cover $G$. In this paper, we show that if $G$ is in the range ${SL}_n\left(q\right)\le G\le {GL}_n\left(q\right)$ for $n>2$, then $n∕{\pi }^2<\gamma \left(G\right)\le \left(n+1\right)∕2$. This result complements recent work of Bubboloni, Praeger and Spiga on symmetric and alternating groups. We give various alternative bounds and derive explicit formulas for $\gamma \left(G\right)$ in some cases.

A composition of an integer constrained to have decreasing then increasing parts is called concave. We prove that the generating function for the number of concave compositions, denoted $v\left(q\right)$, is a mixed mock modular form in a more general sense than is typically used.

We relate $v\left(q\right)$ to generating functions studied in connection with “Moonshine of the Mathieu group” and the smallest parts of a partition. We demonstrate this connection in four different ways. We use the elliptic and modular properties of Appell sums as well as $q$-series manipulations and holomorphic projection.

As an application of the modularity results, we give an asymptotic expansion for the number of concave compositions of $n$. For comparison, we give an asymptotic expansion for the number of concave compositions of $n$ with strictly decreasing and increasing parts, the generating function of which is related to a false theta function rather than a mock theta function.

Abramovich, Corti and Vistoli have studied modular compactifications of stacks of curves equipped with abelian level structures arising as substacks of the stack of twisted stable maps into the classifying stack of a finite group, provided the order of the group is invertible on the base scheme. Recently Abramovich, Olsson and Vistoli extended the notion of twisted stable maps to allow arbitrary base schemes, where the target is a tame stack, not necessarily Deligne–Mumford. We use this to extend the results of Abramovich, Corti and Vistoli to the case of elliptic curves with level structures over arbitrary base schemes; we prove that we recover the compactified Katz–Mazur regular models, with a natural moduli interpretation in terms of level structures on Picard schemes of twisted curves. Additionally, we study the interactions of the different such moduli stacks contained in a stack of twisted stable maps in characteristics dividing the level.

Let $\Gamma$ be a group of order $mp$ where $p$ is prime and $p>m$. We give a strategy to enumerate the regular subgroups of $Perm\left(\Gamma \right)$ normalized by the left representation $\lambda \left(\Gamma \right)$ of $\Gamma$. These regular subgroups are in one-to-one correspondence with the Hopf Galois structures on Galois field extensions $L∕K$ with $\Gamma =Gal\left(L∕K\right)$. We prove that every such regular subgroup is contained in the normalizer in $Perm\left(\Gamma \right)$ of the $p$-Sylow subgroup of $\lambda \left(\Gamma \right)$. This normalizer has an affine representation that makes feasible the explicit determination of regular subgroups in many cases. We illustrate our approach with a number of examples, including the cases of groups whose order is the product of two distinct primes and groups of order $p\left(p-1\right)$, where $p$ is a “safe prime”. These cases were previously studied by N. Byott and L. Childs, respectively.

We prove new inequalities concerning Brauer’s $k\left(B\right)$-conjecture and Olsson’s conjecture by generalizing old results. After that, we obtain the invariants for $2$-blocks of finite groups with certain bicyclic defect groups. Here, a bicyclic group is a product of two cyclic subgroups. This provides an application for the classification of the corresponding fusion systems in a previous paper. To some extent, this generalizes previously known cases with defect groups of types $D_{2^n}\times C_{2^m}$, $Q_{2^n}\times C_{2^m}$ and $D_{2^n}\ast C_{2^m}$. As a consequence, we prove Alperin’s weight conjecture and other conjectures for several new infinite families of nonnilpotent blocks. We also prove Brauer’s $k\left(B\right)$-conjecture and Olsson’s conjecture for the $2$-blocks of defect at most $5$. This completes results from a previous paper. The $k\left(B\right)$-conjecture is also verified for defect groups with a cyclic subgroup of index at most $4$. Finally, we consider Olsson’s conjecture for certain $3$-blocks.

Deodhar introduced his decomposition of partial flag varieties as a tool for understanding Kazhdan–Lusztig polynomials. The Deodhar decomposition of the Grassmannian is also useful in the context of soliton solutions to the KP equation, as shown by Kodama and the second author. Deodhar components ${\mathcal{ℛ}}_D$ of the Grassmannian are in bijection with certain tableaux $D$ called Go-diagrams, and each component is isomorphic to ${\left({\mathbb{K}}^{\ast }\right)}^a\times {\left(\mathbb{K}\right)}^b$ for some nonnegative integers $a$ and $b$.

Our main result is an explicit parametrization of each Deodhar component in the Grassmannian in terms of networks. More specifically, from a Go-diagram $D$ we construct a weighted network $N_D$ and its weight matrix $W_D$, whose entries enumerate directed paths in $N_D$. By letting the weights in the network vary over $\mathbb{K}$ or ${\mathbb{K}}^{\ast }$ as appropriate, one gets a parametrization of the Deodhar component ${\mathcal{ℛ}}_D$. One application of such a parametrization is that one may immediately determine which Plücker coordinates are vanishing and nonvanishing, by using the Lindström–Gessel–Viennot lemma. We also give a (minimal) characterization of each Deodhar component in terms of Plücker coordinates. A main tool for us is the work of Marsh and Rietsch [Represent. Theory 8 (2004), 212–242] on Deodhar components in the flag variety.

Let $X$ be a projective normal toric variety and $T_0$ a rank-$1$ subtorus of the defining torus $T$ of $X$. We show that the normalization of the Chow quotient $X∕∕T_0$, in the sense of Kapranov, Sturmfels, and Zelevinsky, coarsely represents the moduli space of stable log maps to $X$ with discrete data given by $T_0\subset X$. We also obtain similar results when $T_0\to T$ is a homomorphism that is not necessarily an embedding.

Recently, Bhargava and others have proved very striking results about the average size of Selmer groups of Jacobians of algebraic curves over $\mathbb{Q}$ as these curves are varied through certain natural families. Their methods center around the idea of counting integral points in coregular representations, whose rational orbits can be shown to be related to Galois cohomology classes for the Jacobians of these algebraic curves.

In this paper we construct for each simply laced Dynkin diagram a coregular representation $\left(G,V\right)$ and a family of algebraic curves over the geometric quotient $V∕∕G$. We show that the arithmetic of the Jacobians of these curves is related to the arithmetic of the rational orbits of $G$. In the case of type $A_2$, we recover the correspondence between orbits and Galois cohomology classes used by Birch and Swinnerton-Dyer and later by Bhargava and Shankar in their works concerning the 2-Selmer groups of elliptic curves over $\mathbb{Q}$.