22 citations · 51 across the 22 of their papers we have counts for
28 papers
On Vojta's harder implication, height inequalities for admissible pairs, points of bounded degree and Deligne-Mumford stacks
Nathan Grieve
We expand on the concept of \emph{admissible pairs} from \cite{Levin:GCD} and explore its relation with the main Diophantine arithmetic inequalities, with discriminant term and for…
Seshadri slope $\K$-semistability for the blow-up of projective space along a linear subspace
Nathan Grieve
We study a particular instance of the frame work from \cite{Grieve:CM:Line:Slope:Stab}, building on earlier work of Arezzo-et-al \cite{Arezzo:DellaVedova:LaNave}, Ross and Thomas \…
Effective calculation of local Weil functions via presentations of Cartier divisors
Nathan Grieve
We address the question of effectivity for calculation of local Weil functions from the viewpoint of presentations of Cartier divisors. This builds on the approach of Bombieri and…
CM-line bundles and slope $\K$-semistability for big and nef line bundles along subschemes
Nathan Grieve
We apply the theory of the Chow-Mumford line bundle as developed by Arezzo-et-al and build on earlier key insights of Paul and Tian (see \cite{Arezzo:DellaVedova:LaNave} and the re…
About the Bohr-Sommerfeld polytope and the multiplicative -eigenvalue problem for the trinion
Nathan Grieve
We study in detail the work of Jeffrey and Weitsman, especially \cite{Jeffrey:Weitsman:1992} and \cite{Jeffrey:Weitsman:1994}. In particular, we describe the manner in which the in…
Trigonal Canonically Fibered Surfaces
Houari Benammar Ammar, Xi Chen, Nathan Grieve
We fix some gaps of a proof of Xiao's conjecture on canonically fibered surfaces of relative genus 5 by the second author. Our argument simplifies the original proof and gives a mu…
Abstract simplicial complexes in {\tt Macaulay2}
Nathan Grieve
{\tt AbstractSimplicialComplexes.m2} is a computer algebra package written for the computer algebra system {\tt Macaulay2} \cite{M2}. It provides new infrastructure to work with ab…
On qualitative aspects of the quantitative subspace theorem
Nathan Grieve
We deduce Diophantine arithmetic inequalities for big linear systems and with respect to finite extensions of number fields. Our starting point is the Parametric Subspace Theorem,…
Harder-Narasimhan polygons and Laws of Large Numbers
Nathan Grieve
We build on the recent techniques of Codogni and Patakfalvi, from \cite{Codogni:Patakfalvi:2021}, which were used to establish theorems about semi-positivity of the Chow Mumford li…
Numerical contraction for orbifold surfaces
Nathan Grieve
We study singularities and Artin's contraction theorem for orbifold surfaces. Our main result has a consequence which is in the direction of the birational Minimal Model Program fo…
Wedderburn components, the index theorem and continuous Castelnuovo-Mumford regularity for semihomogeneous vector bundles
Nathan Grieve
We study the property of \emph{continuous Castelnuovo-Mumford regularity}, for semihomogeneous vector bundles over a given Abelian variety, which was formulated in \cite{Kuronya:Mu…
On arithmetic inequalities for points of bounded degree
Nathan Grieve
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…
Generalized GCD for toric Fano varieties
Nathan Grieve
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 tog…
On Duistermaat-Heckman measure for filtered linear series
Nathan Grieve
We revisit work of S. Boucksom, C. Favre, and M. Jonsson (J. Algebraic Geom. \textbf{18} (2009), no. 2, 279--308); Boucksom and H. Chen (Compos. Math. \textbf{147} (2011), no. 4, 1…
Greatest common divisors with moving targets and consequences for linear recurrence sequences
Nathan Grieve, Julie Tzu-Yueh Wang
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 multivariabl…
Divisorial instability and Vojta's Main Conjecture for -Fano varieties
Nathan Grieve
We study Diophantine arithmetic properties of birational divisors in conjunction with concepts that surround -stability for Fano varieties. There is also an interpretat…
On cubic torsors, biextensions and Severi-Brauer varieties over Abelian varieties
Nathan Grieve
We study the homogeneous irreducible Severi-Brauer varieties over an Abelian variety . Such objects were classified by Brion, \cite{Bri}. Here we interpret that result within th…
On arithmetic general theorems for polarized varieties
Nathan Grieve
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…
Expectations, Concave Transforms, Chow weights, and Roth's theorem for varieties
Nathan Grieve
We explain how complexity of rational points on projective varieties can be interpreted via the theories of Chow forms and Okounkov bodies. Precisely, we study discrete measures on…
On the Kodaira dimension of maximal orders
Nathan Grieve, Colin Ingalls
Let $\kk$ be an algebraically closed field of characteristic zero and $\KK$ a finitely generated field over $\kk$. Let be a central simple $\KK$-algebra, a normal projectiv…
The software package SpectralSequences
Adam Boocher, Nathan Grieve, Eloísa Grifo
We describe the computer algebra software package SpectralSequences for the computer algebra system Macaulay2. This package implements many data types, objects and algorithms which…
Comments related to infinite wedge representations
Nathan Grieve
We study the infinite wedge representation and show how it is related to the universal extension of the loop algebra of a complex semi-simple Lie algebra . We also…
Reduced norms and the Riemann-Roch theorem for Abelian varieties
Nathan Grieve
We explain how the Riemann-Roch theorem for divisors on an abelian variety is related to the reduced norms of the Wedderburn components of the $\mathb…
Diophantine approximation constants for varieties over function fields
Nathan Grieve
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 a…