We give an upper bound on the volume $\operatorname {vol}(P^*)$ of a polytope $P^*$ dual to a d-dimensional lattice polytope P with exactly one interior lattice point in each dimension d. This bound, expressed in terms of the Sylvester sequence, is sharp and achieved by the dual to a particular reflexive simplex. Our result implies a sharp upper bound on the volume of a d-dimensional reflexive polytope. Translated into toric geometry, this gives a sharp upper bound on the anti-canonical degree $(-K_X)^d$ of a d-dimensional Fano toric variety X with at worst canonical singularities.

We construct an increasing ${\it\omega}$ -sequence $\langle \boldsymbol{a}_{n}\rangle$ of Turing degrees which forms an initial segment of the Turing degrees, and such that each $\boldsymbol{a}_{n+1}$ is diagonally nonrecursive relative to $\boldsymbol{a}_{n}$ . It follows that the DNR principle of reverse mathematics does not imply the existence of Turing incomparable degrees.

We give conditions for a uniruled variety of dimension at least 2 to be nonsolid. This study provides further evidence to a conjecture by Abban and Okada on the solidity of Fano 3-folds. To complement our results we write explicit birational links from Fano 3-folds of high codimension embedded in weighted projective spaces.

Let A be a finite set with , let n be a positive integer, and let $A^n$ denote the discrete $n\text {-dimensional}$ hypercube (that is, $A^n$ is the Cartesian product of n many copies of A). Given a family $\langle D_t:t\in A^n\rangle $ of measurable events in a probability space (a stochastic process), what structural information can be obtained assuming that the events $\langle D_t:t\in A^n\rangle $ are not behaving as if they were independent? We obtain an answer to this problem (in a strong quantitative sense) subject to a mild ‘stationarity’ condition. Our result has a number of combinatorial consequences, including a new (and the most informative so far) proof of the density Hales-Jewett theorem.

We prove an asymptotic expansion of the second moment of the central values of the $\mathrm {GL}(n)\times \mathrm {GL}(n)$ Rankin–Selberg L-functions $L(1/2,\pi \otimes \pi _0)$ for a fixed cuspidal automorphic representation $\pi _0$ over the family of $\pi $ with analytic conductors bounded by a quantity that is tending to infinity. Our proof uses the integral representations of the L-functions, period with regularised Eisenstein series and the invariance properties of the analytic newvectors.

Let X be a smooth projective variety of dimension $n\geq 2$ and $G\cong \mathbf {Z}^{n-1}$ a free abelian group of automorphisms of X over $\overline {\mathbf {Q}}$. Suppose that G is of positive entropy. We construct a canonical height function $\widehat {h}_G$ associated with G, corresponding to a nef and big $\mathbf {R}$-divisor, satisfying the Northcott property. By characterizing the zero locus of $\widehat {h}_G$, we prove the Kawaguchi–Silverman conjecture for each element of G. As for other applications, we determine the height counting function for non-periodic points and show that X satisfies potential density.

A new reciprocity formula for Dirichlet L-functions associated to an arbitrary primitive Dirichlet character of prime modulus q is established. We find an identity relating the fourth moment of individual Dirichlet L-functions in the t-aspect to the cubic moment of central L-values of Hecke–Maaß newforms of level at most $q^{2}$ and primitive central character $\psi ^{2}$ averaged over all primitive nonquadratic characters $\psi $ modulo q. Our formula can be thought of as a reverse version of recent work of Petrow–Young. Direct corollaries involve a variant of Iwaniec’s short interval fourth moment bound and the twelfth moment bound for Dirichlet L-functions, which generalise work of Jutila and Heath-Brown, respectively. This work traverses an intersection of classical analytic number theory and automorphic forms.

The author finds a limit on the singularities that arise in geometric generic fibers of morphisms between smooth varieties of positive characteristic by studying changes in embedding dimension under inseparable field extensions. This result is then used in the context of the minimal model program to rule out the existence of smooth varieties fibered by certain nonnormal del Pezzo surfaces over bases of small dimension.

The K-theoretic Schur P- and Q-functions $G\hspace {-0.2mm}P_\lambda $ and $G\hspace {-0.2mm}Q_\lambda $ may be concretely defined as weight-generating functions for semistandard shifted set-valued tableaux. These symmetric functions are the shifted analogues of stable Grothendieck polynomials and were introduced by Ikeda and Naruse for applications in geometry. Nakagawa and Naruse specified families of dual K-theoretic Schur P- and Q-functions $g\hspace {-0.1mm}p_\lambda $ and $g\hspace {-0.1mm}q_\lambda $ via a Cauchy identity involving $G\hspace {-0.2mm}P_\lambda $ and $G\hspace {-0.2mm}Q_\lambda $. They conjectured that the dual power series are weight-generating functions for certain shifted plane partitions. We prove this conjecture. We also derive a related generating function formula for the images of $g\hspace {-0.1mm}p_\lambda $ and $g\hspace {-0.1mm}q_\lambda $ under the $\omega $ involution of the ring of symmetric functions. This confirms a conjecture of Chiu and the second author. Using these results, we verify a conjecture of Ikeda and Naruse that the $G\hspace {-0.2mm}Q$-functions are a basis for a ring.

Let $X_4\subset \mathbb {P}^{n+1}$ be a quartic hypersurface of dimension $n\geq 4$ over an infinite field k. We show that if either $X_4$ contains a linear subspace $\Lambda $ of dimension $h\geq \max \{2,\dim (\Lambda \cap \operatorname {\mathrm {Sing}}(X_4))+2\}$ or has double points along a linear subspace of dimension $h\geq 3$ , a smooth k-rational point and is otherwise general, then $X_4$ is unirational over k. This improves previous results by A. Predonzan and J. Harris, B. Mazur and R. Pandharipande for quartics. We also provide a density result for the k-rational points of quartic $3$ -folds with a double plane over a number field, and several unirationality results for quintic hypersurfaces over a $C_r$ field.

Let G be a permutation group on a finite set $\Omega $. The base size of G is the minimal size of a subset of $\Omega $ with trivial pointwise stabiliser in G. In this paper, we extend earlier work of Fawcett by determining the precise base size of every finite primitive permutation group of diagonal type. In particular, this is the first family of primitive groups arising in the O’Nan–Scott theorem for which the exact base size has been computed in all cases. Our methods also allow us to determine all the primitive groups of diagonal type with a unique regular suborbit.

We give new descriptions of the Bruhat order and Demazure products of affine Weyl groups in terms of the weight function of the quantum Bruhat graph. These results can be understood to describe certain closure relations concerning the Iwahori–Bruhat decomposition of an algebraic group. As an application towards affine Deligne–Lusztig varieties, we present a new formula for generic Newton points.

For a finite group $G$, let $d(G)$ denote the minimal number of elements required to generate $G$. In this paper, we prove sharp upper bounds on $d(H)$ whenever $H$ is a maximal subgroup of a finite almost simple group. In particular, we show that $d(H)\leqslant 5$ and that $d(H)\geqslant 4$ if and only if $H$ occurs in a known list. This improves a result of Burness, Liebeck and Shalev. The method involves the theory of crowns in finite groups.

We study a relative variant of Serre’s notion of $G$-complete reducibility for a reductive algebraic group $G$. We let $K$ be a reductive subgroup of $G$, and consider subgroups of $G$ that normalize the identity component $K^{\circ }$. We show that such a subgroup is relatively $G$-completely reducible with respect to $K$ if and only if its image in the automorphism group of $K^{\circ }$ is completely reducible. This allows us to generalize a number of fundamental results from the absolute to the relative setting. We also derive analogous results for Lie subalgebras of the Lie algebra of $G$, as well as ‘rational’ versions over nonalgebraically closed fields.

For $\theta $ a small generic universal stability condition of degree $0$ and A a vector of integers adding up to $-k(2g-2+n)$, the spaces $\overline {\mathcal {M}}_{g,A}^\theta $ constructed in [AP21, HMP+22] are observed to lie inside the space $\textbf {Div}$ of [MW20], and their pullback under $\textbf {Rub} \to \textbf {Div}$ of loc. cit. to be smooth. This provides smooth and modular modifications $\widetilde {\mathcal {M}}_{g,A}^\theta $ of $\overline {\mathcal {M}}_{g,n}$ on which the logarithmic double ramification cycle can be calculated by several methods.

In this paper, we prove that klt singularities are invariant under deformations if the generic fiber is $\mathbb {Q}$ -Gorenstein. We also obtain a similar result for slc singularities. These are generalizations of results of Esnault-Viehweg [Math. Ann. 271 (1985), 439–449] and S. Ishii [Math. Ann. 275 (1986), 139–148; Singularities (Iowa City, IA, 1986) Contemporary Mathematics, vol. 90 (American Mathematical Society, Providence, RI, 1989), 135–145].

We show that there exists a cubic threefold defined by an invertible polynomial that, when quotiented by the maximal diagonal symmetry group, has a derived category that does not have a full exceptional collection consisting of line bundles. This provides a counterexample to a conjecture of Lekili and Ueda.

The goal of this paper is to describe certain nonlinear topological obstructions for the existence of first-order smoothings of mildly singular Calabi–Yau varieties of dimension at least $4$. For nodal Calabi–Yau threefolds, a necessary and sufficient linear topological condition for the existence of a first-order smoothing was first given in [Fri86]. Subsequently, Rollenske–Thomas [RT09] generalized this picture to nodal Calabi–Yau varieties of odd dimension by finding a necessary nonlinear topological condition for the existence of a first-order smoothing. In a complementary direction, in [FL22a], the linear necessary and sufficient conditions of [Fri86] were extended to Calabi–Yau varieties in every dimension with $1$-liminal singularities (which are exactly the ordinary double points in dimension $3$ but not in higher dimensions). In this paper, we give a common formulation of all of these previous results by establishing analogues of the nonlinear topological conditions of [RT09] for Calabi–Yau varieties with weighted homogeneous k-liminal hypersurface singularities, a broad class of singularities that includes ordinary double points in odd dimensions.

Let n be a nonnegative integer. For each composition $\alpha $ of n, Berg, Bergeron, Saliola, Serrano and Zabrocki introduced a cyclic indecomposable $H_n(0)$ -module $\mathcal {V}_{\alpha }$ with a dual immaculate quasisymmetric function as the image of the quasisymmetric characteristic. In this paper, we study $\mathcal {V}_{\alpha }$ s from the homological viewpoint. To be precise, we construct a minimal projective presentation of $\mathcal {V}_{\alpha }$ and a minimal injective presentation of $\mathcal {V}_{\alpha }$ as well. Using them, we compute $\mathrm {Ext}^1_{H_n(0)}(\mathcal {V}_{\alpha }, \mathbf {F}_{\beta })$ and $\mathrm {Ext}^1_{H_n(0)}( \mathbf {F}_{\beta }, \mathcal {V}_{\alpha })$ , where $\mathbf {F}_{\beta }$ is the simple $H_n(0)$ -module attached to a composition $\beta $ of n. We also compute $\mathrm {Ext}_{H_n(0)}^i(\mathcal {V}_{\alpha },\mathcal {V}_{\beta })$ when $i=0,1$ and $\beta \le _l \alpha $ , where $\le _l$ represents the lexicographic order on compositions.

Let X be a smooth proper variety over a field k and suppose that the degree map ${\mathrm {CH}}_0(X \otimes _k K) \to \mathbb {Z}$ is isomorphic for any field extension $K/k$. We show that $G(\operatorname {Spec} k) \to G(X)$ is an isomorphism for any $\mathbb {P}^1$-invariant Nisnevich sheaf with transfers G. This generalises a result of Binda, Rülling and Saito that proves the same conclusion for reciprocity sheaves. We also give a direct proof of the fact that the unramified logarithmic Hodge–Witt cohomology is a $\mathbb {P}^1$-invariant Nisnevich sheaf with transfers.