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John Lesieutre

Publications and source records attributed to John Lesieutre.

17 recordsLinked to original sources
arXiv2025

The volume of a divisor and cusp excursions of geodesics in hyperbolic manifolds

We give a complete description of the behavior of the volume function at the boundary of the pseudoeffective cone of certain Calabi-Yau complete intersections known as Wehler N-folds. We find that the volume function exhibits a pathological behavior when N>=3, we obtain examples of a pseudoeffective R-divisor D for which the volume of D+sA, with s small and A ample, oscillates between two powers of s, and we deduce the sharp regularity of this function answering a question of Lazarsfeld. We also show that h^0(X,[mD]+A) displays a similar oscillatory behavior as m increases, showing that several notions of numerical dimensions of D do not agree and disproving a conjecture of Fujino. We accomplish this by relating the behavior of the volume function along a segment to the visits of a corresponding hyperbolic geodesics to the cusps of a hyperbolic manifold.

math.AG↗
arXiv2020

Tri-Coble surfaces and their automorphisms

We construct some positive entropy automorphisms of rational surfaces with no periodic curves. The surfaces in question, which we term tri-Coble surfaces, are blow-ups of the projective plane at 12 points which have contractions down to three different Coble surfaces. The automorphisms arise as compositions of lifts of Bertini involutions from certain degree 1 weak del Pezzo surfaces.

math.AG↗
arXiv2019

Higher arithmetic degrees of dominant rational self-maps

Suppose that $f \colon X \dashrightarrow X$ is a dominant rational self-map of a smooth projective variety defined over ${\overline{\mathbf Q}}$. Kawaguchi and Silverman conjectured that if $P \in X({\overline{\mathbf Q}})$ is a point with well-defined forward orbit, then the growth rate of the height along the orbit exists, and coincides with the first dynamical degree $λ_1(f)$ of $f$ if the orbit of $P$ is Zariski dense in $X$. In this note, we extend the Kawaguchi-Silverman conjecture to the setting of orbits of higher-dimensional subvarieties of $X$. We begin by defining a set of arithmetic degrees of $f$, independent of the choice of cycle, and we then develop the theory of arithmetic degrees in parallel to existing results for dynamical degrees. We formulate several conjectures governing these higher arithmetic degrees, relating them to dynamical degrees.

math.NT↗
arXiv2019

Notions of numerical Iitaka dimension do not coincide

Let $X$ be a smooth projective variety. The Iitaka dimension of a divisor $D$ is an important invariant, but it does not only depend on the numerical class of $D$. However, there are several definitions of ``numerical Iitaka dimension'', depending only on the numerical class. In this note, we show that there exists a pseuodoeffective $\mathbb R$-divisor for which these invariants take different values. The key is the construction of an example of a pseudoeffective $\mathbb R$-divisor $D_+$ for which $h^0(X,\lfloor m D_+ \rfloor+A)$ is bounded above and below by multiples of $m^{3/2}$ for any sufficiently ample $A$.

math.AG↗
arXiv2018

A rational map with infinitely many points of distinct arithmetic degrees

Let $f \colon X \dashrightarrow X$ be a dominant rational self-map of a smooth projective variety defined over $\overline{\mathbb Q}$. For each point $P\in X(\overline{\mathbb Q})$ whose forward $f$-orbit is well-defined, Silverman introduced the arithmetic degree $α_f(P)$, which measures the growth rate of the heights of the points $f^n(P)$. Kawaguchi and Silverman conjectured that $α_f(P)$ is well-defined and that, as $P$ varies, the set of values obtained by $α_f(P)$ is finite. Based on constructions of Bedford--Kim and McMullen, we give a counterexample to this conjecture when $X=\mathbb P^4$.

math.AG↗
arXiv2018

Canonical heights on hyper-Kähler varieties and the Kawaguchi-Silverman conjecture

The Kawaguchi--Silverman conjecture predicts that if $f\colon X \dashrightarrow X$ is a dominant rational-self map of a projective variety over $\overline{\mathbb{Q}}$, and $P$ is a $\overline{\mathbb{Q}}$-point of $X$ with Zariski-dense orbit, then the dynamical and arithmetic degrees of $f$ coincide: $λ_1(f) = α_f(P)$. We prove this conjecture in several higher-dimensional settings, including all endomorphisms of non-uniruled smooth projective threefolds with degree larger than $1$, and all endomorphisms of hyper-Kähler varieties in any dimension. In the latter case, we construct a canonical height function associated to any automorphism $f\colon X \to X$ of a hyper-Kähler variety defined over $\overline{\mathbb{Q}}$.

math.AG↗
arXiv2016

Dynamical Mordell-Lang and Automorphisms of Blow-ups

We show that if $ϕ: X \to X$ is an automorphism of a smooth projective variety and $D \subset X$ is an irreducible divisor for which the set of $d$ in $D$ with $ϕ^n(d)$ in $D$ for some nonzero $n$ is not Zariski dense, then $(X, ϕ)$ admits an equivariant rational fibration to a curve. As a consequence, we show that certain blowups (e.g. blowups in high codimension) do not alter the finiteness of $\textrm{Aut}(X)$, extending results of Bayraktar-Cantat. We also generalize results of Arnol'd on the growth of multiplicities of the intersection of a variety with the iterates of some other variety under an automorphism. These results follow from a non-reduced analogue of the dynamical Mordell-Lang conjecture. Namely, let $ϕ: X \to X$ be an étale endomorphism of a smooth projective variety $X$ over a field $k$ of characteristic zero. We show that if $Y$ and $Z$ are two closed subschemes of $X$, then the set $A_ϕ(Y,Z) = \{n : ϕ^n(Y) \subseteq Z\}$ is the union of a finite set and finitely many residue classes, whose modulus is bounded in terms of the geometry of $Y$.

math.AG↗
arXiv2016

Effective cones of cycles on blow-ups of projective space

In this paper, we study the cones of higher codimension (pseudo)effective cycles on point blow-ups of projective space. We determine bounds on the number of points for which these cones are generated by the classes of linear cycles, and for which these cones are finitely generated. Surprisingly, we discover that for (very) general points, the higher codimension cones behave better than the cones of divisors. For example, for the blow-up $X_r^n$ of $\mathbb P^n$, $n>4$, at $r$ very general points, the cone of divisors is not finitely generated as soon as $r> n+3$, whereas the cone of curves is generated by the classes of lines if $r \leq 2^n$. In fact, if $X_r^n$ is a Mori Dream Space then all the effective cones of cycles on $X_r^n$ are finitely generated.

math.AG↗
arXiv2015

A few questions about curves on surfaces

In this note we address the following kind of question: let X be a smooth, irreducible, projective surface and D a divisor on X$satisfying some sort of positivity hypothesis, then is there some multiple of D depending only on X which is effective or movable? We describe some examples, discuss some conjectures and prove some results that suggest that the answer should in general be negative, unless one puts some really strong hypotheses either on D or on X.

math.AG↗
arXiv2015

Some constraints on positive entropy automorphisms of smooth threefolds

Suppose that $X$ is a smooth, projective threefold over $\mathbb C$ and that $ϕ: X \to X$ is an automorphism of positive entropy. We show that one of the following must hold, after replacing $ϕ$ by an iterate: i) the canonical class of $X$ is numerically trivial; ii) $ϕ$ is imprimitive; iii) $ϕ$ is not dynamically minimal. As a consequence, we show that if a smooth threefold $M$ does not admit a primitive automorphism of positive entropy, then no variety constructed by a sequence of smooth blow-ups of $M$ can admit a primitive automorphism of positive entropy. In explaining why the method does not apply to threefolds with terminal singularities, we exhibit a non-uniruled, terminal threefold $X$ with infinitely many $K_X$-negative extremal rays on $NE(X)$.

math.AG↗
arXiv2015

A pathology of asymptotic multiplicity in the relative setting

We point out an example of a projective family $π: X \to S$, a $π$-pseudoeffective divisor $D$ on $X$, and a subvariety $V \subset X$ for which the asymptotic multiplicity $σ_V(D;X/S)$ is infinite. This shows that the divisorial Zariski decomposition is not always defined for pseudoeffective divisors in the relative setting.

math.AG↗
arXiv2014

Curves disjoint from a nef divisor

On a projective surface it is well-known that the set of curves orthogonal to a nef line bundle is either finite or uncountable. We show that this dichotomy fails in higher dimension by constructing a nef line bundle on a threefold which is trivial on countably infinitely many curves. This answers a question of Totaro. As a pleasant corollary, we exhibit a quasi-projective variety with only a countably infinite set of complete, positive-dimensional subvarieties.

math.AG↗
arXiv2014

The diminished base locus is not always closed

We exhibit a pseudoeffective R-divisor D_λon the blow-up of P^3 at nine very general points which lies in the closed movable cone and has negative intersections with a set of curves whose union is Zariski dense. It follows that the diminished base locus B_-(D_λ) = \bigcup_{A ample}} B(D_λ+A) is not closed and that D_λdoes not admit a Zariski decomposition in even a very weak sense. By a similar method, we construct an R-divisor on the family of blow-ups of P^2 at ten distinct points, which is nef on a very general fiber but fails to be nef over countably many prime divisors in the base.

math.AG↗
arXiv2013

Derived-equivalent rational threefolds

We describe an infinite set of smooth projective threefolds that have equivalent derived categories but are not isomorphic, contrary to a conjecture of Kawamata. These arise as blow-ups of $\mathbb P^3$ at various configurations of 8 points, which are related by Cremona transformations.

math.AG↗
arXiv2008

Analytic continuation of Dirichlet series with almost periodic coefficients

We prove that an ordinary Dirichlet series with coefficients a(n)=g(n b) has an abscissa of convergence 0 if g is an odd 1-periodic, real-analytic function and b is Diophantine. We also show that if g is odd and has bounded variation and b is of bounded Diophantine type r>1, then the abscissa of convergence is smaller or equal than 1-1/r. Using a polylogarithm expansion, we prove that if g is odd and real analytic and b is Diophantine, then the ordinary Dirichlet series has an analytic continuation to the entire complex plane.

math.CV↗