Seshadri constants, diophantine approximation, and Roth’s theorem for arbitrary varieties
By David McKinnon, Mike Roth. 2013
74 citations recorded by the source corpus.
In this paper, we associate an invariant $$\alpha _{x}(L)$$αx(L) to an algebraic point $$x$$x on an algebraic variety $$X$$X with an ample line bundle $$L$$L. The invariant $$\alpha $$α measures how well $$x$$x can be approximated by rational points on $$X$$X, with respect to the height function associated to $$L$$L. We show that this invariant is closely related to the Seshadri constant $$\epsilon _{x}(L)$$ϵx(L) measuring local positivity of $$L$$L at $$x$$x, and in particular that Roth’s theorem on $$\mathbb {P}^1$$P1 generalizes as an inequality between these two invariants valid for arbitrary projective varieties.
Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/3d43ef23ac8e3edeaa1dc55d422fd6866351eac4
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Citations in the corpus
- Thresholds, valuations, and K-stability (2017, background): “ Li [Li17], and Y. Liu [Liu18]. In the notation of [Fuj19a], T(ord E) = ˝(E) and S(ord E) = ˝(E) vol(L) 1j(E): The invariant S(ord ˘), for ˘ 2X a regular closed point, also plays an important role in [MR15] and was used in unpublished work of P. Salberger from 2006. Proposition 3.11. If Lis ample an”
- A non-Archimedean approach to K-stability (2018, background): “Second, the last equality shows that S ( x ) agrees with the expected order of vanishing in [BKMS16, BlJ17], see also [MR15].”
- A variational approach to the Yau–Tian–Donaldson conjecture (2015, background): “In particular, if x ∈ X , then S L (ord x ) coincides with the invariant considered in [MR15, § 4].”