# Generalized GCD for toric Fano varieties

@article{Grieve2020GeneralizedGF, title={Generalized GCD for toric Fano varieties}, author={Nathan Mark Grieve}, journal={Acta Arithmetica}, year={2020} }

We study the greatest common divisor problem for torus invariant blowing-up morphisms of nonsingular toric Fano varieties. Our main result applies the theory of Okounkov bodies together with an arithmetic form of Cartan's Second Main theorem, which has been established by Ru and Vojta. It also builds on Silverman's geometric concept of greatest common divisor. As a special case of our results, we deduce a bound for the generalized greatest common divisor of pairs of nonzero algebraic numbers.

## 6 Citations

An arithmetic general theorem for points of bounded degree.

- Mathematics
- 2019

We study rational points of bounded degree on polarized projective varieties. To do so, we refine further the filtration construction and subspace theorem approach, for the study of integral points,…

On arithmetic inequalities for points of bounded degree

- MathematicsResearch in Number Theory
- 2020

We study algebraic points of bounded degree on polarized projective varieties. To do so, we refine further the filtration construction and Subspace Theorem approach, for the study of integral points,…

Greatest common divisors for polynomials in almost units and applications to linear recurrence sequences

- Mathematics
- 2021

We bound the greatest common divisor of two coprime multivariable polynomials evaluated at algebraic numbers, generalizing work of Levin, and going towards conjectured inequalities of Silverman and…

VERTICES OF THE HARDER AND NARASIMHAN POLYGONS AND THE LAWS OF LARGE NUMBERS

- MathematicsCanadian Mathematical Bulletin
- 2022

We build on the recent techniques of Codogni and Patakfalvi, from [4], which were used to establish theorems about semi-positivity of the Chow Mumford line bundles for families of K-semistable Fano…

On Duistermaat-Heckman measure for filtered linear series

- Mathematics
- 2019

We revisit work of S. Boucksom, C. Favre, and M. Jonsson (J. Algebraic Geom. 18 (2009), no. 2, 279–308); Boucksom and H. Chen (Compos. Math. 147 (2011), no. 4, 1205– 1229); and S. Boucksom, A.…

Greatest common divisors with moving targets and consequences for linear recurrence sequences

- Mathematics
- 2020

We establish consequences of the moving form of Schmidt’s Subspace Theorem. Indeed, we obtain inequalities that bound the logarithmic greatest common divisor of moving multivariable polynomials…

## References

SHOWING 1-10 OF 20 REFERENCES

Greatest common divisors and Vojta’s conjecture for blowups of algebraic tori

- MathematicsInventiones mathematicae
- 2018

We give results and inequalities bounding the greatest common divisor of multivariable polynomials evaluated at S-unit arguments, generalizing to an arbitrary number of variables results of…

On arithmetic general theorems for polarized varieties.

- Mathematics
- 2017

We apply Schmidt's Subspace Theorem to establish two Arithmetic General Theorems. For example, we establish an Arithmetic General Theorem for projective varieties over number and function fields.…

Introduction to Toric Varieties.

- Mathematics
- 1993

Toric varieties are algebraic varieties arising from elementary geometric and combinatorial objects such as convex polytopes in Euclidean space with vertices on lattice points. Since many algebraic…

Heights in Diophantine Geometry

- Mathematics
- 2006

I. Heights II. Weil heights III. Linear tori IV. Small points V. The unit equation VI. Roth's theorem VII. The subspace theorem VIII. Abelian varieties IX. Neron-Tate heights X. The Mordell-Weil…

Convex Bodies Associated to Linear Series

- Mathematics
- 2008

In his work on log-concavity of multiplicities, Okounkov showed in passing that one could associate a convex body to a linear series on a projective variety, and then use convex geometry to study…

A subspace theorem for subvarieties

- Mathematics
- 2017

In this paper, we establish a height inequality, in terms of an (ample) line bundle, for a sum of subschemes located in `-sub-general position in an algebraic variety, which extends the main result…

An Effective Schmidt’s Subspace Theorem for Projective Varieties Over Function Fields

- Mathematics
- 2012

We deduce an effective version of Schmidt’s subspace theorem on a smooth projective variety over a function field of characteristic zero for divisors of coming from hypersurfaces in .

Scalar Curvature and Stability of Toric Varieties

- Mathematics
- 2002

We define a stability condition for a polarised algebraic variety and state a conjecture relating this to the existence of a Kahler metric of constant scalar curvature. The main result of the paper…

Arithmetic distance functions and height functions in diophantine geometry

- Mathematics
- 1987

In virtually all areas of Diophantine Geometry, the theory of height functions plays a crucial role. This theory associates to each (Cartier) divisor D on a projective variety V a function ho mapping…

Expectations, Concave Transforms, Chow weights, and Roth's theorem for varieties over number fields

- Mathematics
- 2017

We study discrete measures on filtered linear series and build on work of Boucksom-Chen, Boucksom-et-al and Ferretti. Specifically, we establish connections between the expectations of such measures…