We give conditions for the monodromy group of a Hurwitz space over the configuration space of branch points to be the full alternating or symmetric group on the degree. Specializing the resulting coverings suggests the existence of many number fields with surprisingly little ramification —for example, the existence of infinitely many $A_m$ or $S_m$ number fields unramified away from $\left\{2,3,5\right\}$.

The Adams operators ${\Psi }_n$ on a Hopf algebra $H$ are the convolution powers of the identity of $H$. They are also called Hopf powers or Sweedler powers. We study the Adams operators when $H$ is graded connected. The main result is a complete description of the characteristic polynomial — both eigenvalues and their multiplicities — for the action of the operator ${\Psi }_n$ on each homogeneous component of $H$. The eigenvalues are powers of $n$. The multiplicities are independent of $n$, and in fact only depend on the dimension sequence of $H$. These results apply in particular to the antipode of $H$, as the case $n=-1$. We obtain closed forms for the generating function of the sequence of traces of the Adams operators. In the case of the antipode, the generating function bears a particularly simple relationship to the one for the dimension sequence. In the case where $H$ is cofree, we give an alternative description for the characteristic polynomial and the trace of the antipode in terms of certain palindromic words. We discuss parallel results that hold for Hopf monoids in species and for $q$-Hopf algebras.

On a special line bundle $L$ on a projective curve $C$ we introduce a geometric condition called $\left({\Delta }_q\right)$. When $L=K_C$, this condition implies $gon\left(C\right)\ge q+2$. For an arbitrary special $L$, we show that $\left({\Delta }_3\right)$ implies that $L$ has the well-known property $\left(M_3\right)$, generalising a similar result proved by Voisin in the case $L=K_C$.

We prove that the double covers of the alternating and symmetric groups are determined by their complex group algebras. To be more precise, let $n\ge 5$ be an integer, $G$ a finite group, and let ${\hat{A}}_n$ and ${\hat{S}}_n^{\pm }$ denote the double covers of $A_n$ and $S_n$, respectively. We prove that $ℂG\cong ℂ{\hat{A}}_n$ if and only if $G\cong {\hat{A}}_n$, and $ℂG\cong ℂ{\hat{S}}_n^{+}\cong ℂ{\hat{S}}_n^{-}$ if and only if $G\cong {\hat{S}}_n^{+}$ or ${\hat{S}}_n^{-}$. This in particular completes the proof of a conjecture proposed by the second and fourth authors that every finite quasisimple group is determined uniquely up to isomorphism by the structure of its complex group algebra. The known results on prime power degrees and relatively small degrees of irreducible (linear and projective) representations of the symmetric and alternating groups together with the classification of finite simple groups play an essential role in the proofs.

Let $D_{m,n}^r$ and $P_{m,n}^r$ denote the subschemes of ${\mathbb{P}}^{mn-1}$ given by the $r\times r$ determinants (respectively the $r\times r$ permanents) of an $m\times n$ matrix of indeterminates. In this paper, we study the geometry of the Fano schemes $F_k\left(D_{m,n}^r\right)$ and $F_k\left(P_{m,n}^r\right)$ parametrizing the $k$-dimensional planes in ${\mathbb{P}}^{mn-1}$ lying on $D_{m,n}^r$ and $P_{m,n}^r$, respectively. We prove results characterizing which of these Fano schemes are smooth, irreducible, and connected; and we give examples showing that they need not be reduced. We show that $F_1\left(D_{n,n}^n\right)$ always has the expected dimension, and we describe its components exactly. Finally, we give a detailed study of the Fano schemes of $k$-planes on the $3\times 3$ determinantal and permanental hypersurfaces.

Let $X$ be an isotropic Grassmannian of type $B$, $C$, or $D$. In this paper we calculate $K$-theoretic Pieri-type triple intersection numbers for $X$: that is, the sheaf Euler characteristic of the triple intersection of two arbitrary Schubert varieties and a special Schubert variety in general position. We do this by determining explicit equations for the projected Richardson variety corresponding to the two arbitrary Schubert varieties, and show that it is a complete intersection in projective space. The $K$-theoretic Pieri coefficients are alternating sums of these triple intersection numbers, and we hope they will lead to positive Pieri formulas for isotropic Grassmannians.

We prove the basepoint-free theorem for big line bundles on a three-dimensional log canonical projective pair defined over the algebraic closure of a finite field. This theorem is not valid for any other algebraically closed field.

Let $G$ be a finite group and let $T\left(G\right)$ be the abelian group of equivalence classes of endotrivial $kG$-modules, where $k$ is an algebraically closed field of characteristic $p$. We determine, in terms of the structure of $G$, the kernel of the restriction map from $T\left(G\right)$ to $T\left(S\right)$, where $S$ is a Sylow $p$-subgroup of $G$, in the case when $S$ is abelian. This provides a classification of all torsion endotrivial $kG$-modules in that case.