John Lesieutre

Papers:

"Tri-Coble surfaces and their automorphisms". (arXiv:2003.01799) abstract±
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.

"Higher arithmetic degrees of dominant rational self-maps", submitted. (arXiv:1906.11188) abstract±
Suppose that f : X ⇢ X is a dominant rational self-map of a smooth projective variety defined over Q. Kawaguchi and Silverman conjectured that if P ∈ 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 λ₁(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.

"Notions of numerical Iitaka dimension do not coincide", to appear in J. Algebraic Geometry. (arXiv:1904.10832) abstract±
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 R-divisor for which these invariants take different values. The key is the construction of an example of a pseudoeffective R-divisor D₊ for which h⁰(X,⌊mD₊⌋+A) is bounded above and below by multiples of m^3/2 for any sufficiently ample A.

"A rational map with infinitely many points of distinct arithmetic degrees" (with M. Satriano), to appear in Ergodic Theory and Dynamical Systems. (arXiv:1809.00047) abstract±
Let f : X ⇢ X be a dominant rational self-map of a smooth projective variety defined over Q. For each point P ∈ 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 fⁿ(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 = P⁴.

"Canonical heights on hyper-Kähler varieties and the Kawaguchi–Silverman conjecture" (with M. Satriano), to appear in Int. Math. Res. Not.. (arXiv:1802.07388) abstract±
The Kawaguchi–Silverman conjecture predicts that if f : X ⇢ X is a dominant rational-self map of a projective variety over Q, and P is a Q-point of X with Zariski-dense orbit, then the dynamical and arithmetic degrees of f coincide: λ₁(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 : X → X of a hyper-Kähler variety defined over Q.

"A projective variety with discrete, non-finitely generated automorphism group", Inventiones Math 212 (2018), no. 1, 189–211. (arXiv:1609.06391) (sage, output) abstract±
We construct a projective variety with discrete, non-finitely generated automorphism group. As an application, we show that there exists a complex projective variety with infinitely many non-isomorphic real forms.

"Dynamical Mordell-Lang and automorphisms of blow-ups" (with D. Litt), Algebraic Geometry 6 (2019), no. 1, 1–25. (arXiv:1604.08216) abstract±
We show that if φ : X → X is an automorphism of a smooth projective variety and D ⊂ 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 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 → 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 : φⁿ(Y) ⊂ Z} is the union of a finite set and finitely many residue classes, whose modulus is bounded in terms of the geometry of Y.

"Log Fano structures and Cox rings of blow-ups of products of projective spaces" (with J. Park), Proc. Amer. Math. Soc. 145 (2017), no. 10, 4201–4209. (arXiv:1604.07140) abstract±
The aim of this paper is twofold. Firstly, we determine which blow-ups of products of projective spaces at general points are varieties of Fano type, and give boundary divisors making these spaces log Fano pairs. Secondly, we describe generators of the Cox rings of some cases.

"Effective cones of cycles on blow-ups of projective space" (with I. Coskun and J.C. Ottem), Algebra & Number Theory 10-9 (2016). (arXiv:1603.04808) abstract±
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 Xⁿ_r of Pⁿ, 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 ≤ 2ⁿ. In fact, if Xⁿ_r is a Mori Dream Space then all the effective cones of cycles on Xⁿ_r are finitely generated.

"A few questions about curves on surfaces" (with Ciliberto, Knutsen, Lozovanu, Miranda, Mustopa, and Testa), Rend. Circ. Mat. Palermo (2016), 1–10. (arXiv:1511.06618) abstract±
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.

"Some constraints on positive entropy automorphisms of smooth threefolds", to appear in Ann. Sci. École Norm. Sup.. (arXiv:1503.07834) abstract±
Suppose that X is a smooth, projective threefold over C and that φ : X → 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).

"A pathology of asymptotic multiplicity in the relative setting", Math. Res. Lett. 23 (2016), no. 5, 1433–1451. (arXiv:1502.03019) abstract±
We point out an example of a projective family π : X → S, a π-pseudoeffective divisor D on X, and a subvariety V ⊂ 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.

"Curves disjoint from a nef divisor" (with J.C. Ottem), Michigan Math. J. 65 (2016), 321–332.(arXiv:1410.4467) abstract±
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 an effective, 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.

"Derived-equivalent rational threefolds", Int. Math. Res. Not. (2015) 6011–6020. (journal / arXiv:1311.0056) abstract±
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 P³ at various configurations of 8 points, which are related by Cremona transformations.

"The diminished base locus is not always closed", Compositio Math. 150 (2014), no. 10, 1729–1741. (journal / arXiv:1212.3738) abstract±
We exhibit a pseudoeffective R-divisor D_∞ on the blow-up of P³ 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_∞) = ∪_{A ample} B(D_∞+A) is not closed and that D_∞ does 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 P² 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.