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Arc lifting for the Nash manifold
[article]

2020
*
arXiv
*
pre-print

In addition

arXiv:1103.5723v10
fatcat:amrlgr3q4bfsfa6xj3gcih5xrm
*a**theorem*about valuations is proven. ...*A**theorem*of Functorial Affinization of Nash's manifold is proven here giving necessary and sufficient conditions to lift*a*holomorphic arc to the smooth locus of the Nash manifold. ...*A*coherent sheaf D on*a*quasiprojective scheme is 'coprimary' (generalizing the*definition*given in Algebre Local, Multiplicities' [20] ) if there is*a**prime**ideal*sheaf P so that if there is any*prime*...##
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Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta-functions, notes on the work of Shinichi Mochizuki

2015
*
European Journal of Mathematics
*

In

doi:10.1007/s40879-015-0066-0
fatcat:xrpw4vw74jd5ved7srzh3jnksy
*a*series of preprints released several years ago, Mochizuki introduced*a*vast collection of new ideas and methods related to arithmetic deformation. ... Abstract These notes survey the main ideas, concepts and objects of the work by Shinichi Mochizuki on inter-*universal*Teichmüller theory (Inter-*universal*Teichmüller theory I-IV, 2012-2015) which might ... Work on this text was partially supported by*University*of Nottingham and EPSRC programme grant EP/M024830. ...##
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Arithmetic dynamics on smooth cubic surfaces
[article]

2014
*
arXiv
*
pre-print

We present results describing the sets of K and K̅-periodic points of the system, and give

arXiv:1307.3205v2
fatcat:mnrlapvnvza2vjxrfmtya32p5a
*a*necessary and sufficient condition for*a*dynamical local-global property called strong residual periodicity. ... Finally, we give*a*dynamical result relating to the Mordell--Weil problem on cubic surfaces. ... the maximal*ideal*in O p . • κ p is the residue field of the*prime*p (i.e. ...##
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Newton-Okounkov bodies and Segre classes
[article]

2018
*
arXiv
*
pre-print

Given

arXiv:1809.07344v1
fatcat:v453zp4ci5dyvbphqil2g6r2rm
*a*homogeneous*ideal*in*a*polynomial ring over C, we adapt the*construction*of Newton-Okounkov bodies to obtain*a*convex subset of Euclidean space such that*a*suitable integral over this set computes ... the Segre zeta function of the*ideal*. ... The*indeterminacies*of ϕ s are*resolved*by the closure Γ s of the graph of ϕ s ,*a*subvariety of dimension n of P n × P Ns . ...##
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The jumping coefficients of non-Q-Gorenstein multiplier ideals

2016
*
Journal of Algebra
*

We also obtain that outside of W, the multiplier

doi:10.1016/j.jalgebra.2015.11.024
fatcat:ssisajbozvc4tpr2x43ol2xhyi
*ideal*reduces to the test*ideal*modulo sufficiently large*primes*p ≫ 0. ... Let*a*⊂ O_X be*a*coherent*ideal*sheaf on*a*normal complex variety X, and let c > 0 be*a*real number. ... Furthermore, it answers*a*question of Rossi [Ros68, p. 72] asking for*a**universal*property of his*construction*.*Definition*6.1 (Resolution of*a*sheaf) . ...##
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Indeterministic finite-precision physics and intuitionistic mathematics
[article]

2021
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arXiv
*
pre-print

Next, we investigate whether we can use intuitionistic mathematics to mathematically express the idea of

arXiv:2108.05735v1
fatcat:yrpuc6jysndxxpknoo6cx2tctu
*finite*precision of quantities, arriving at the cautious conclusion that, as far as we can see, ... Finally, we outline another approach to formalising*finite*-precision quantities in classical mechanics, which is inspired by the intuitionistic approach to the continuum but uses classical mathematics. ... sum of two*primes*. ...##
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Modular curves and Mordell-Weil torsion in F-theory

2020
*
Journal of High Energy Physics
*

We prove this by explicitly

doi:10.1007/jhep04(2020)103
fatcat:ftimrypun5gbjj5h6gfwo4fkpe
*constructing*and studying maps to*a*modular curve whose existence is predicted by*a**universal*property. ... Article funded by SCOAP 3 . https://doi.org/10.1007/JHEP04(2020)103 JHEP04(2020)103 C*Universal*curves for Z 2 × Z 2m and Z m × Z 2m 42 C.1 Set-up 42 C.2*Construction*43 C.3 Z 2 × Z 4 44 7 For*a*more detailed ... Let p ⊂ R be*a**prime**ideal*and I ⊂ R an*ideal*. The order of vanishing of I at p is the largest integer m such that I ⊂ p m .*Definition*5.6. Let φ = p q ∈ K, with p, q ∈ R relatively*prime*. ...##
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Saturation, seminormalization and homeomorphisms of algebraic varieties
[article]

2022
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arXiv
*
pre-print

We address the question under which conditions

arXiv:2203.09967v1
fatcat:g4d7ofkwibepxm776ptkadw4hi
*a*bijective morphism between algebraic varieties over an algebraically closed field of characteristic zero is an isomorphism. ... Our two answers involve*a*study of seminormalization and saturation for morphisms between algebraic varieties, together with an interpretation in terms of continuous rational functions on the closed points ... In order to study the saturation, we need to understand better the relation between*prime**ideals*in*A*and B and*prime**ideals*in B ⊗*A*B. ...##
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Functorial affinization of Nash's manifold
[article]

2020
*
arXiv
*
pre-print

When there are infinitely many nonzero terms the sum should be taken formally or p-adically for

arXiv:1004.2234v15
fatcat:ialjsgwly5avpl4xt25dcvyeqe
*a**prime*divisor p of n+2. There are*finitely*many nonzero terms if and only if C. D=0. ... If M is projective, singular foliations on M such that K+(n+1)H is*a**finitely*-generated divisor of Iitaka dimension one are completely*resolvable*, where K is the canonical divisor of the foliation and ... Then W is an algebraic variety and Y 0 → M is not only*a*proper map, it is actualy the*universal*solution of*resolving*the*indeterminacies*of the rational map M− → W, ie it is the closure of the 'graph ...##
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A uniform treatment of Grothendieck's localization problem
[article]

2021
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arXiv
*
pre-print

To answer Grothendieck's problem, we provide

arXiv:2004.06737v2
fatcat:3kwsw6ikgff35pbukmhoihbmry
*a*general framework that gives*a*uniform treatment of previously known cases of this problem, and also solves this problem in new cases, namely for weak normality ... Our proof of this statement reduces to*a*purely local question known as Grothendieck's localization problem. ...*prime**ideal*p ⊆*A*; and (b) We*Definition*2.7 [ILO14, Exp. ...##
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Unpolarized Shafarevich conjectures for hyper-Kähler varieties
[article]

2022
*
arXiv
*
pre-print

The Shafarevich conjecture/problem is about the

arXiv:2203.10391v1
fatcat:ty6z2foggnfcldg6u6jtgf5a5y
*finiteness*of isomorphism classes of*a*family of varieties defined over*a*number field with good reduction outside*a**finite*collection of places. ... In*a*similar fashion, generalizing*a*result of Orr and Skorobogatov on K3 surfaces, we prove the*finiteness*of geometric isomorphism classes of hyper-K\"ahler varieties of CM type in*a*given deformation ... 8.3 and Lemma*A*.3. ...##
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Modular Curves and Mordell-Weil Torsion in F-theory
[article]

2019
*
arXiv
*
pre-print

We prove this by explicitly

arXiv:1910.04095v2
fatcat:p3italufbveb7gp5l2sjfzmojq
*constructing*and studying maps to*a*modular curve whose existence is predicted by*a**universal*property. ... curves to explain the minimal singularities that appear on an elliptic fibration with prescribed torsion, and to determine the degree of the fundamental line bundle (hence the Kodaira dimension) of the*universal*... Let m ⊂ R be*a*maximal*ideal*and I ⊂ R an*ideal*. The order of vanishing of I at m is the largest integer m such that I ⊂ m m .*Definition*5.6. Let φ = p q ∈ K, with p, q ∈ R relatively*prime*. ...##
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The Ando-Hopkins-Rezk orientation is surjective
[article]

2020
*
arXiv
*
pre-print

We do so by

arXiv:1911.10534v3
fatcat:dhablqdo3bghtkrjv6a5wfojom
*constructing*an E_1-ring B and*a*map B →MString such that the composite B →MString→tmf is surjective on homotopy. ... The*universal*property of*A*from Remark 2.5 therefore produces an E 1 -map*A*→ BP. ... In this section, we show that this conjecture follows from*Theorem*1.2. We begin by recalling the*definition*of the E ∞ -ring spectra M j r .*Definition*6.1. ...##
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Terminal valuations and the Nash problem

2015
*
Inventiones Mathematicae
*

In dimension two, this result gives

doi:10.1007/s00222-015-0597-5
fatcat:hcy2k2mqt5fwhg2kn2sgtxi5s4
*a*new proof of the*theorem*of Fern\'andez de Bobadilla and Pe Pereira stating that, for surfaces, the Nash map is*a*bijection. ... Terminal valuations are defined in the sense of the minimal model program, as those valuations given by the exceptional divisors on*a*minimal model over X. ... The*indeterminacies*of S • (Z ′ ) • can be*resolved*by*a*minimal sequence Z • → S • of blow-ups of maximal*ideals*. ...##
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The stable Adams conjecture and higher associative structures on Moore spectra
[article]

2021
*
arXiv
*
pre-print

Our proof

arXiv:1803.11014v3
fatcat:pb5b6zypvbeolebyqt4cy337yy
*constructs**a*canonical null-homotopy of the stable J-homomorphism composed with*a*virtual Adams operation, by applying the K-theory functor to*a*multi-natural transformation. ... More precisely, for any odd*prime*p, we show that M_p(k) admits*a*Thomified 𝔸_n-structure if and only if n < p^k. We also prove*a*weaker result for p=2. ... Thus, invoking*Theorem*2.12 we produce*a*canonical null-homotopy which*resolves**Theorem*1.5 -the unreduced stable Adams conjecture.*Definition*5.2. ...
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