Related papers: A conjecture of Ax and degenerations of Fano varie…

We investigate a relationship between nondegeneracy of a simple abelian variety $A$ over an algebraic closure of $\mb{Q}$ and of its reduction $A_0$. We prove that under some assumptions, nondegeneracy of $A$ implies nondegeneracy of $A_0$.

Mahowald's conjecture arose as part of a program attempting to view chromatic phenomena in stable homotopy theory through the lens of the classical Adams spectral sequence. The conjecture predicts the existence of nonzero classes in the…

For every natural number $m$, the existentially closed models of the theory of fields with $m$ commuting derivations can be given a first-order geometric characterization in several ways. In particular, the theory of these differential…

In the large rank limit, for any nonexceptional affine algebra, the graded branching multiplicities known as one-dimensional sums, are conjectured to have a simple relationship with those of type A, which are known as generalized Kostka…

We study congruences of lines $X_\omega$ defined by a sufficiently general choice of an alternating 3-form $\omega$ in $n+1$ dimensions, as Fano manifolds of index $3$ and dimension $n-1$. These congruences include the…

We prove the generalised Mukai conjecture for $\mathbb{Q}$-factorial spherical Fano varieties. In this case, a stronger inequality holds featuring an extra term - the minimum absolute complexity of a log Calabi-Yau pair - which measures how…

Marker and Steinhorn shown that given two models $M\prec N$ of an o-minimal theory, if all 1-types over $M$ realized in $N$ are definable, then all types over $M$ realized in $N$ are definable. In this article we characterize pairs of…

We show that an earlier conjecture of the author, on diophantine approximation of rational points on varieties, implies the ``abc conjecture'' of Masser and Oesterl'e. In fact, a weak form of the former conjecture is sufficient, involving…

Let $X$ be a hyperk\"ahler variety. Voisin has conjectured that the classes of Lagrangian constant cycle subvarieties in the Chow ring of $X$ should lie in a subring injecting into cohomology. We study this conjecture for the Fano variety…

We classify codimension two analytic submanifolds X of projective space having the property that any line through a general point p having contact to order two with X at p automatically has contact to order three. We give applications to…

We prove the stable degeneration conjecture of log Fano fibration germs formulated by Sun-Zhang. Precisely, we introduce the $\mathbf{H}$-invariant for filtrations over a log Fano fibration germ, and show that there exists a unique…

In this talk, I report on three theorems concerning algebraic varieties over a field of characteristic $p>0$. a) over a finite field of cardinal $q$, two proper smooth varieties which are geometrically birational have the same number of…

We define the nef complexity of a projective variety $X$. This invariant compares $\dim X+\rho(X)$ with the sum of the coefficients of nef partitions of $-K_X$. We prove that the nef complexity is non-negative and it is zero precisely for…

We prove that every globally $F$-regular variety is log Fano. In other words, if a prime characteristic variety $X$ is globally $F$-regular, then it admits an effective $\bQ$-divisor $\Delta$ such that $-K_X - \Delta$ is ample and $(X,…

We obtain several finiteness results for the unramified cohomology of function fields of algebraic varieties defined over fields of type (F'_m), a class that includes algebraically closed fields, finite fields, local fields, and some higher…

We introduce a notion of Mathieu-Zhao subspaces of vertex algebras. Among other things, we show that for a vertex algebra $V$ and its subspace $M$ that contains $C_2(V)$, $M$ is a Mathieu-Zhao subspace of $V$ if and only if the quotient…

We study varieties defined by parameterizing polynomials of derivatives through a computational algebro-geometric approach, especially relying on Combinatorial Nullstellensatz and Noether normalization. We establish that these polynomials…

Let $\pi: X \to Y$ be a morphism of projective varieties and suppose that $\alpha$ is a pseudo-effective numerical cycle class satisfying $\pi_*\alpha = 0$. A conjecture of Debarre, Jiang, and Voisin predicts that $\alpha$ is a limit of…

We propose a conjectural list of Fano manifolds of Picard number $1$ with pseudoeffective normalized tangent bundles, which we prove in various situations by relating it to the complete divisibility conjecture of Russo and Zak on varieties…

Bloch's conjecture for a surface $X$ over an algebraically closed field $k$ states that every homologically trivial correspondence $\Gamma $ acts as 0 on the Albanese kernel $T(X_{\Omega})$, where $\Omega $ is a universal domain containing…