We prove the Green–Lazarsfeld secant conjecture [Green and Lazarsfeld, On the projective normality of complete linear series on an algebraic curve, Invent. Math. 83 (1986), 73–90; Conjecture (3.4)] for extremal line bundles on curves of arbitrary gonality, subject to explicit genericity assumptions.

We correct an error in our paper ‘Connected components of affine Deligne–Lusztig varieties in mixed characteristic’.

In this paper we study holomorphic Legendrian curves in the standard holomorphic contact structure on $\mathbb{C}^{2n+1}$ for any $n\in \mathbb{N}$ . We provide several approximation and desingularization results which enable us to prove general existence theorems, settling some of the open problems in the subject. In particular, we show that every open Riemann surface $M$ admits a proper holomorphic Legendrian embedding $M{\hookrightarrow}\mathbb{C}^{2n+1}$ , and we prove that for every compact bordered Riemann surface $M={M\unicode[STIX]{x0030A}}\,\cup \,bM$ there exists a topological embedding $M{\hookrightarrow}\mathbb{C}^{2n+1}$ whose restriction to the interior is a complete holomorphic Legendrian embedding ${M\unicode[STIX]{x0030A}}{\hookrightarrow}\mathbb{C}^{2n+1}$ . As a consequence, we infer that every complex contact manifold $W$ carries relatively compact holomorphic Legendrian curves, normalized by any given bordered Riemann surface, which are complete with respect to any Riemannian metric on $W$ .

We prove analogs of the Bezout and the Bernstein–Kushnirenko–Khovanskii theorems for systems of algebraic differential conditions over differentially closed fields. Namely, given a system of algebraic conditions on the first $l$ derivatives of an $n$ -tuple of functions, which admits finitely many solutions, we show that the number of solutions is bounded by an appropriate constant (depending singly-exponentially on $n$ and $l$ ) times the volume of the Newton polytope of the set of conditions. This improves a doubly-exponential estimate due to Hrushovski and Pillay. We illustrate the application of our estimates in two diophantine contexts: to counting transcendental lattice points on algebraic subvarieties of semi-abelian varieties, following Hrushovski and Pillay; and to counting the number of intersections between isogeny classes of elliptic curves and algebraic varieties, following Freitag and Scanlon. In both cases we obtain bounds which are singly-exponential (improving the known doubly-exponential bounds) and which exhibit the natural asymptotic growth with respect to the degrees of the equations involved.

Let $k$ be a finite extension of $\mathbb{Q}_{p}$ , let ${\mathcal{G}}$ be an absolutely simple split reductive group over  $k$ , and let $K$ be a maximal unramified extension of $k$ . To each point in the Bruhat–Tits building of ${\mathcal{G}}_{K}$ , Moy and Prasad have attached a filtration of ${\mathcal{G}}(K)$ by bounded subgroups. In this paper we give necessary and sufficient conditions for the dual of the first Moy–Prasad filtration quotient to contain stable vectors for the action of the reductive quotient. Our work extends earlier results by Reeder and Yu, who gave a classification in the case when $p$ is sufficiently large. By passing to a finite unramified extension of $k$ if necessary, we obtain new supercuspidal representations of ${\mathcal{G}}(k)$ .

Extending our previous work on eigenvalues of closed surfaces and work of Otal and Rosas, we show that a complete Riemannian surface $S$ of finite type and Euler characteristic $\unicode[STIX]{x1D712}(S)<0$ has at most $-\unicode[STIX]{x1D712}(S)$ small eigenvalues.