An official website of the United States government
Let 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 $$\unicode[STIX]{x1D6FC}_{f}(P)$$ , which measures the growth rate of the heights of the points $$f^{n}(P)$$ . Kawaguchi and Silverman conjectured that $$\unicode[STIX]{x1D6FC}_{f}(P)$$ is well defined and that, as $$P$$ varies, the set of values obtained by $$\unicode[STIX]{x1D6FC}_{f}(P)$$ is finite. Based on constructions by Bedford and Kim and by McMullen, we give a counterexample to this conjecture when $$X=\mathbb{P}^{4}$$ .
more »
« less
Heights on stacks and a generalized Batyrev–Manin–Malle conjecture
Abstract We define a notion of height for rational points with respect to a vector bundle on a proper algebraic stack with finite diagonal over a global field, which generalizes the usual notion for rational points on projective varieties. We explain how to compute this height for various stacks of interest (for instance: classifying stacks of finite groups, symmetric products of varieties, moduli stacks of abelian varieties, weighted projective spaces). In many cases, our uniform definition reproduces ways already in use for measuring the complexity of rational points, while in others it is something new. Finally, we formulate a conjecture about the number of rational points of bounded height (in our sense) on a stack $$\mathcal {X}$$ , which specializes to the Batyrev–Manin conjecture when $$\mathcal {X}$$ is a scheme and to Malle’s conjecture when $$\mathcal {X}$$ is the classifying stack of a finite group.
more »
« less
- Award ID(s):
- 2001200
- PAR ID:
- 10425468
- Date Published:
- Journal Name:
- Forum of Mathematics, Sigma
- Volume:
- 11
- ISSN:
- 2050-5094
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
null (Ed.)Abstract Projective varieties with ample cotangent bundle satisfy many notions of hyperbolicity, and one goal of this paper is to discuss generalizations to quasi-projective varieties. A major hurdle is that the naive generalization is false—the log cotangent bundle is never ample. Instead, we define a notion called almost ample that roughly asks that it is as positive as possible. We show that all subvarieties of a quasi-projective variety with almost ample log cotangent bundle are of log general type. In addition, if one assumes globally generated then we obtain that such varieties contain finitely many integral points. In another direction, we show that the Lang–Vojta conjecture implies the number of stably integral points on curves of log general type, and surfaces of log general type with almost ample log cotangent sheaf are uniformly bounded.more » « less
-
Abstract We study integral points on varieties with infinite étale fundamental groups. More precisely, for a number field $$F$$ and $X/F$ a smooth projective variety, we prove that for any geometrically Galois cover $$\varphi \colon Y \to X$$ of degree at least $$2\dim (X)^{2}$$, there exists an ample line bundle $$\mathscr{L}$$ on $$Y$$ such that for a general member $$D$$ of the complete linear system $$|\mathscr{L}|$$, $$D$$ is geometrically irreducible and any set of $$\varphi (D)$$-integral points on $$X$$ is finite. We apply this result to varieties with infinite étale fundamental group to give new examples of irreducible, ample divisors on varieties for which finiteness of integral points is provable.more » « less
-
Resolvent degree is an invariant measuring the complexity of algebraic and geometric phenomena, including the complexity of finite groups. To date, the resolvent degree of a finite simple group G has only been investigated when G is a cyclic group; an alternating group; a simple factor of a Weyl group of type E6, E7, or E8; or PSL(2,F7). In this paper, we establish upper bounds on the resolvent degrees of the sporadic groups by using the invariant theory of their projective representations. To do so, we introduce the notion of (weak) RDk≤d-versality, which we connect to the existence of “special points” on varieties.more » « less
-
Abstract Cluster varieties come in pairs: for any {\mathcal{X}}cluster variety there is an associated Fock–Goncharov dual {\mathcal{A}}cluster variety. On the other hand, in the context of mirror symmetry, associated with any log Calabi–Yau variety is its mirror dual, which can be constructed using the enumerative geometry of rational curves in the framework of the Gross–Siebert program. In this paper we bridge the theory of cluster varieties with the algebro-geometric framework of Gross–Siebert mirror symmetry. Particularly, we show that the mirror to the {\mathcal{X}}cluster variety is a degeneration of the Fock–Goncharov dual {\mathcal{A}}cluster varietyand vice versa. To do this, we investigate how the cluster scattering diagram of Gross, Hacking, Keel and Kontsevich compares with the canonical scattering diagram defined by Gross and Siebert to construct mirror duals in arbitrary dimensions. Consequently, we derive an enumerative interpretation of the cluster scattering diagram. Along the way, we prove the Frobenius structure conjecture for a class of log Calabi–Yau varieties obtained as blow-ups of toric varieties.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript
- Journal Article:
- https://doi.org/10.1017/fms.2023.5
