Algebraic Geometry and Arithmetic Curves

By Qing Liu. 2002

1,394 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/899a5e04fcb72c3c2b2c615ea21effeabe7099e9

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Citations in the corpus

Quadratic Chabauty and rational points, I: p-adic heights (2016, neutral): “x2 = αx1 + β, y2 = γy1 + δx1 + ǫ (see [34], Corollary 7.”
On different notions of tameness in arithmetic geometry (2008, methodology): “ank d = dimS X. Using Lemma 3.5, we ﬁnd a wildly ramiﬁed geometric discrete rank 1 valuation v on k(X). Choose any normal compactiﬁcation X¯ of X. As Y → X is étale, the center of v lies in X¯ rX. By [Liu] §8, ex. 3.14, after blowing up X¯ in centers contained in X¯ rX and ﬁnally normalizing, we ﬁnd”
Uniform bounds for the number of rational points on curves of small Mordell–Weil rank (2015, neutral): “Then, since X is regular, the adjunction formula [Liu02, Theorem 9.1.36] gives O(V )|ΓX = −∆(f) ≔ ∑ v ∑ e=vw (f(w)− f(v))(v).”
The index of an algebraic variety (2012, background): “Proof (1) As we blow-up a regular scheme along the regular center Spec k(x0), we have X̃ regular and E1 Pdk(x0) ([39], 8.”
Specialization of linear systems from curves to graphs (2007, background): “It is well known that the natural map from X (K ) = X(R) to the smooth locus of Xk(k) is surjective (see for example [Liu 2002, Proposition 10.1.40(b)]); by (A.4), this implies that the natural maps Div(X)→ Div(G) and Div0(X)→ Div0(G) are surjective.”
Local-global principles for torsors over arithmetic curves (2011, background): “Since ∞ defines an ample divisor on P1k, the forward implication of the proposition now follows from the fact that the inverse image of an ample divisor under a finite surjective morphism is ample ([Liu02], Remark 5.3.9).”
Hypersurfaces in projective schemes and a moving lemma (2014, background): “Since S is a Dedekind scheme and X → S is flat with X integral, we find that all fibers are equidimensional of dimension dim Xη = 1 ([44], 4.”
The Grothendieck ring of varieties and piecewise isomorphisms (2010, background): “By Lüroth’s Theorem (see for instance [26], Corollary 7.”
Estimation of geodesic tortuosity and constrictivity in stationary random closed sets (2018, background): “Indeed, by Liu (2002, Proposition 5.5) the intersection of any two non-identical non-singular curves is of dimension 0 and therefore consists of a finite union of points (Liu, 2002, Proposition 4.9, Lemma 5.11). Moreover, with probability 1, the four curves associated to the intersection of triplets”
Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell–Weil rank (2013, methodology): “By [Liu02, Corollary 9.3.26], g ≥ 2 implies that Γ ·W ≥ 0.”
On the Brauer group of a surface (2005, neutral): “, [7], 9.”
Zhang’s Conjecture and the Effective Bogomolov Conjecture over function fields (2009, neutral): “We refer to [24 ]a nd [ 16 ] for the proofs of the mentioned facts.”
Lectures on K3 Surfaces (2016, background): “Popular sources for elliptic surfaces include, among many others, [32, 131, 184, 373, 412].”