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We are looking for a formulation of Manin's real multiplication question in higher rank. This question comports, in our point of view, at least two steps: 1. a formalisation of the linear algebra side of the story in terms of morphisms of…
This note discusses some intriguing connections between height bounds on complex K-semistable Fano varieties X and Peyre's conjectural formula for the density of rational points on X. Relations to an upper bound for the smallest rational…
We study moduli spaces of stable objects in the Kuznetsov components of Fano threefolds. We prove a general non-emptiness criterion for moduli spaces, which applies to the cases of prime Fano threefolds of index $1$, degree $10 \leq d \leq…
In this paper we establish an asymptotic formula for the number of rational points of bounded anticanonical height which lie on a certain Zariski dense subset of the biprojective hypersurface \begin{align*} x_1y_1^2+...+x_sy_s^2 = 0…
We study the asymptotic growth of the number of rational points of bounded height on smooth projective split toric varieties with Picard rank 2 over number fields, with respect to Arakelov height functions associated with big metrized line…
We prove divisorial canonicity of Fano double hypersurfaces of general position.
The purpose of this paper is to compute the degree of irrationality of hypersurfaces of sufficiently high degree in various Fano varieties: quadrics, Grassmannians, products of projective space, cubic threefolds, cubic fourfolds, and…
Let $B$ be a smooth projective curve and let $\pi: \mathcal{X} \to B$ be a smooth integral model of a geometrically integral Fano variety over $K(B)$. Geometric Manin's Conjecture predicts the structure of the irreducible components $M…
We describe a method that we believe may be foundational for a comprehensive theory of generalised Turan problems. The cornerstone of our approach is a quasirandom counting lemma for quasirandom hypergraphs, which extends the standard…
We introduce some invariants of Fano varieties and propose a Mukai-type conjecture which characterizes the product of projective spaces. Moreover, we prove that the Ambro--Kawamata effective non-vanishing conjecture implies the Mukai-type…
We investigate the first-order correction in the homogenization of linear parabolic equations with random coefficients. In dimension $3$ and higher and for coefficients having a finite range of dependence, we prove a pointwise version of…
We study Fano varieties endowed with a faithful action of a symmetric group, as well as analogous results for Calabi--Yau varieties, and log terminal singularities. We show the existence of a constant $m(n)$, so that every symmetric group…
In this paper, we give a uniform upper bound on the rational points of bounded height provided by conics in a cubic surface. For this target, we give a generalized version of the global determinant method of Salberger by Arakelov geometry.
We prove Manin's conjecture concerning the distribution of rational points of bounded height, and its refinement by Peyre, for wonderful compactifications of semi-simple algebraic groups over number fields. The proof proceeds via the study…
In this paper, we prove the conic version of YTD conjecture on log Fano manifolds.
By a heuristic argument, we relate two conjectures. One is a version of Manin's conjecture about the distribution of rational points on a Fano variety. We concern specific singular Fano varieties, namely quotients of projective spaces by…
Let $X$ be a cubic fourfold that has only simple singularities and does not contain a plane. We prove that the Fano variety of lines on $X$ has the same analytic type of singularity as the Hilbert scheme of two points on a surface with only…
We prove the thin set version of Manin's conjecture for the chordal (or: determinantal) cubic fourfold, which is the secant variety of the Veronese surface. We reduce this counting problem to a result of Schmidt for quadratic points in the…
In this paper, we study the explicit geometry of threefolds, in particular, Fano varieties. We find an explicitly computable positive integer $N$, such that all but a bounded family of Fano threefolds have $N$-complements. This result has…
We prove Kawaguchi-Silverman conjecture (KSC) and Shibata's conjecture on ample canonical heights for endomorphisms on several classes of algebraic varieties including varieties of Fano type and projective toric varieties. We also prove KSC…