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Nathan Grieve

Publications and source records attributed to Nathan Grieve.

28 records · Page 2Linked to original sources
arXiv2021

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

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 evaluated at moving $S$-unit arguments. In doing so, we complement recent work of Levin. As an additional application, we obtain results that pertain to the greatest common divisor problem for algebraic linear recurrence sequences. These observations are motivated by previous related works of Corvaja-Zannier, Levin and others.

math.NT↗
arXiv2020

Divisorial instability and Vojta's Main Conjecture for $\mathbb{Q}$-Fano varieties

We study Diophantine arithmetic properties of birational divisors in conjunction with concepts that surround $\mathrm{K}$-stability for Fano varieties. There is also an interpretation in terms of the barycentres of Newton-Okounkov bodies. Our main results show how the notion of divisorial instability, in the sense of K. Fujita, implies instances of Vojta's Main Conjecture for Fano varieties. A main tool in the proof of these results is an arithmetic form of Cartan's Second Main Theorem that has been obtained by M. Ru and P. Vojta.

math.AG↗
arXiv2019

Generalized GCD for toric Fano varieties

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.

math.AG↗
arXiv2018

Index conditions and cup-product maps on abelian varieties

We study questions surrounding cup-product maps which arise from pairs of non-degenerate line bundles on an abelian variety. Important to our work is Mumford's index theorem which we use to prove that non-degenerate line bundles exhibit positivity analogous to that of ample line bundles. As an application we determine the asymptotic behaviour of families of cup-product maps and prove that vector bundles associated to these families are asymptotically globally generated. To illustrate our results we provide several examples. For instance, we construct families of cup-product problems which result in a zero map on a one dimensional locus. We also prove that the hypothesis of our results can be satisfied, in all possible instances, by a particular class of simple abelian varieties. Finally, we discuss the extent to which Mumford's theta groups are applicable in our more general setting.

math.AG↗
arXiv2018

Refinements to Mumford's theta and adelic theta groups

Let $X$ be an abelian variety defined over an algebraically closed field $k$. We consider theta groups associated to \emph{simple semi-homogenous vector bundles of separable type} on $X$. We determine the structure and representation theory of these groups. In doing so we relate work of Mumford, Mukai, and Umemura. We also consider adelic theta groups associated to line bundles on $X$. After reviewing Mumford's construction of these groups we determine functorial properties which they enjoy and then realize the Neron-Severi group of $X$ as a subgroup of the cohomology group $\H^2(\V(X); k^\times)$.

math.AG↗
arXiv2018

On arithmetic general theorems for polarized varieties

We apply Schmidt's Subspace Theorem to establish Arithmetic General Theorems for projective varieties over number and function fields. Our first result extends an analogous result of M. Ru and P. Vojta. One aspect to its proof makes use of a filtration construction which appears in work of Autissier. Further, we consider work of M. Ru and J. T.-Y. Wang, which pertains to an extension of K. F. Roth's theorem for projective varieties in the sense of D. McKinnon and M. Roth. Motivated by these works, we establish our second Arithmetic General Theorem, namely a form of Roth's theorem for exceptional divisors. Finally, we observe that our results give, within the context of Fano varieties, a sufficient condition for validity of the main inequalities predicted by Vojta.

math.AG↗
arXiv2017

Reduced norms and the Riemann-Roch theorem for Abelian varieties

We explain how the Riemann-Roch theorem for divisors on an abelian variety $A$ is related to the reduced norms of the Wedderburn components of $\operatorname{End}^0(A)$ the $\mathbb{Q}$-endomorphism algebra of $A$. We then describe consequences and examples.

math.AG↗
arXiv2017

Diophantine approximation constants for varieties over function fields

By analogy with the program of McKinnon-Roth, we define and study approximation constants for points of a projective variety X defined over K the function field of an irreducible and non-singular in codimension 1 projective variety defined over an algebraically closed field of characteristic zero. In this setting, we use an effective version of Schmidt's subspace theorem, due to J.T.-Y. Wang, to give a sufficient condition for such approximation constants to be computed on a proper K-subvariety of X. We also indicate how our approximation constants are related to volume functions and Seshadri constants.

math.AG↗
arXiv2016

The software package SpectralSequences

We describe the computer algebra software package SpectralSequences for the computer algebra system Macaulay2. This package implements many data types, objects and algorithms which pertain to, among other things, filtered complexes, spectral sequence pages and maps therein. We illustrate some of the syntax and capabilities of SpectralSequences by way of several examples.

math.AC↗