We present a conceptual and uniform interpretation of the methods of integral representations of $L$-functions (period integrals, Rankin–Selberg integrals). This leads to (i) a way to classify such integrals, based on the classification of certain embeddings of spherical varieties (whenever the latter is available), (ii) a conjecture that would imply a vast generalization of the method, and (iii) an explanation of the phenomenon of “weight factors” in a relative trace formula. We also prove results of independent interest, such as the generalized Cartan decomposition for spherical varieties of split groups over $p$-adic fields (following an argument of Gaitsgory and Nadler).

We prove that the sweeping components of the space of smooth rational curves in a smooth hypersurface of degree $d$ in ${\mathbb{P}}^n$ are not uniruled if $\left(n+1\right)∕2\le d\le n-3$. We also show that for any $e\ge 1$, the space of smooth rational curves of degree $e$ in a general hypersurface of degree $d$ in ${\mathbb{P}}^n$ is not uniruled roughly when $d\ge e\sqrt{n}$.

We define a new symmetry for morphisms of vector bundles, called triality symmetry, and compute Chern class formulas for the degeneracy loci of such morphisms. In an appendix, we show how to canonically associate an octonion algebra bundle to any rank-$2$ vector bundle.

We show that the equation $a^2+b^6=c^n$ has no nontrivial positive integer solutions with $\left(a,b\right)=1$ via a combination of techniques based upon the modularity of Galois representations attached to certain $\mathbb{Q}$-curves, corresponding surjectivity results of Ellenberg for these representations, and extensions of multi-Frey curve arguments of Siksek.

Suppose that closed subschemes $X\subset Y\subset {\mathbb{P}}^N$ differ at finitely many points: when is $Y$ a flat specialization of $X$ union isolated points? Our main result says that this holds if $X$ is a local complete intersection of codimension two and the multiplicity of each embedded point of $Y$ is at most three. We show by example that no hypothesis can be weakened: the conclusion fails for embedded points of multiplicity greater than three, for local complete intersections $X$ of codimension greater than two, and for nonlocal complete intersections of codimension two. As applications, we determine the irreducible components of Hilbert schemes of space curves with high arithmetic genus and show the smoothness of the Hilbert component whose general member is a plane curve union a point in ${\mathbb{P}}^3$.

We define a proper moduli stack classifying covers of curves of prime degree $p$. The objects of this stack are torsors $Y\to \mathcal{X}$ under a finite flat $\mathcal{X}$-group scheme, with $\mathcal{X}$ a twisted curve and $Y$ a stable curve. We also discuss embeddings of finite flat group schemes of order $p$ into affine smooth $1$-dimensional group schemes.

Let $F$ be a field of prime characteristic $p$ and let $B$ be a nonprincipal block of the group algebra $F{S}_r$ of the symmetric group $S_r$. The block component $Lie{\left(r\right)}_B$ of the Lie module $Lie\left(r\right)$ is projective, by a result of Erdmann and Tan, although $Lie\left(r\right)$ itself is projective only when $p\nmid r$. Write $r=p^{m}k$, where $p\nmid k$, and let $S_k^{\ast }$ be the diagonal of a Young subgroup of $S_r$ isomorphic to $S_k\times \cdots \times S_k$. We show that $p^{m}Lie{\left(r\right)}_B\cong {\left(Lie\left(k\right){\uparrow }_{S_k^{\ast }}^{S_r}\right)}_B$. Hence we obtain a formula for the multiplicities of the projective indecomposable modules in a direct sum decomposition of $Lie{\left(r\right)}_B$. Corresponding results are obtained, when $F$ is infinite, for the $r$-th Lie power $L^r\left(E\right)$ of the natural module $E$ for the general linear group ${GL}_n\left(F\right)$.

We reformulate basepoint-free theorems using notions introduced by Shokurov, such as b-divisors and saturation of linear systems. Our formulation is flexible and has some important applications. One of the main purposes of this paper is to prove a generalization of the basepoint-free theorem in Fukuda’s paper “On numerically effective log canonical divisors”.

Using results of the author with Cohen and Nakano, we find examples of Young modules $Y^{\lambda }$ for the symmetric group ${\Sigma }_d$ for which the Tate cohomology ${\hat{H}}^i\left({\Sigma }_d,Y^{\lambda }\right)$ does not vanish identically, but vanishes for approximately $\frac{1}{3}{d}^{3∕2}$ consecutive degrees. We conjecture these vanishing ranges are maximal among all ${\Sigma }_d$-modules with nonvanishing cohomology. The best known upper bound on such vanishing ranges stands at ${\left(d-1\right)}^2$, due to work of Benson, Carlson and Robinson. Particularly striking, and perhaps counterintuitive, is that these Young modules have maximum possible complexity.