Higher connectivity of tropicalizations

We show that the tropicalization of an irreducible d-dimensional variety over a field of characteristic 0 is $$(d-\ell )$$ ( d - ℓ ) -connected through codimension one, where $$\ell $$ ℓ is the dimension of the lineality space of the tropicalization. From this we obtain a higher connectivity result for skeleta of rational polytopes. We also prove a tropical analogue of the Bertini Theorem: the intersection of the tropicalization of an irreducible variety with a generic hyperplane is again the tropicalization of an irreducible variety.

Generic rank of Betti map and unlikely intersections

Let $\mathcal {A} \rightarrow S$ be an abelian scheme over an irreducible variety over $\mathbb {C}$ of relative dimension $g$. For any simply-connected subset $\Delta$ of $S^{\mathrm {an}}$ one can define the Betti map from $\mathcal {A}_{\Delta }$ to $\mathbb {T}^{2g}$, the real torus of dimension $2g$, by identifying each closed fiber of $\mathcal {A}_{\Delta } \rightarrow \Delta$ with $\mathbb {T}^{2g}$ via the Betti homology. Computing the generic rank of the Betti map restricted to a subvariety $X$ of $\mathcal {A}$ is useful to study Diophantine problems, e.g. proving the geometric Bogomolov conjecture over char $0$ and studying the relative Manin–Mumford conjecture. In this paper we give a geometric criterion to detect this rank. As an application we show that it is maximal after taking a large fibered power (if $X$ satisfies some conditions); it is an important step to prove the bound for the number of rational points on curves (Dimitrov et al., Uniformity in Mordell–Lang for Curves, Preprint (2020), arXiv:2001.10276). Another application is to answer a question of André, Corvaja and Zannier and improve a result of Voisin. We also systematically study its link with the relative Manin–Mumford conjecture, reducing the latter to a simpler conjecture. Our tools are functional transcendence and unlikely intersections for mixed Shimura varieties.

The fibration method over real function fields

Let $$\mathbb R(C)$$ R ( C ) be the function field of a smooth, irreducible projective curve over $$\mathbb R$$ R . Let X be a smooth, projective, geometrically irreducible variety equipped with a dominant morphism f onto a smooth projective rational variety with a smooth generic fibre over $$\mathbb R(C)$$ R ( C ) . Assume that the cohomological obstruction introduced by Colliot-Thélène is the only one to the local-global principle for rational points for the smooth fibres of f over $$\mathbb R(C)$$ R ( C ) -valued points. Then we show that the same holds for X, too, by adopting the fibration method similarly to Harpaz–Wittenberg.

On Grothendieck–Serre’s conjecture concerning principal -bundles over reductive group schemes: I

Let$k$be an infinite field. Let$R$be the semi-local ring of a finite family of closed points on a$k$-smooth affine irreducible variety, let$K$be the fraction field of$R$, and let$G$be a reductive simple simply connected$R$-group scheme isotropic over$R$. Our Theorem 1.1 states that for any Noetherian$k$-algebra$A$the kernel of the map$$\begin{eqnarray}H_{\acute{\text{e}}\text{t}}^{1}(R\otimes _{k}A,G)\rightarrow H_{\acute{\text{e}}\text{t}}^{1}(K\otimes _{k}A,G)\end{eqnarray}$$induced by the inclusion of$R$into$K$is trivial. Theorem 1.2 for$A=k$and some other results of the present paper are used significantly in Fedorov and Panin [A proof of Grothendieck–Serre conjecture on principal bundles over a semilocal regular ring containing an infinite field, Preprint (2013),arXiv:1211.2678v2] to prove the Grothendieck–Serre’s conjecture for regular semi-local rings$R$containing an infinite field.

ALGEBRO-GEOMETRIC INVARIANTS OF GROUPS (THE DIMENSION SEQUENCE OF A REPRESENTATION VARIETY)

If G is a finitely generated group and A is an algebraic group, then RA(G) = Hom (G, A) is an algebraic variety. Define the "dimension sequence" of G over A as Pd(RA(G)) = (Nd(RA(G)), …, N0(RA(G))), where Ni(RA(G)) is the number of irreducible components of RA(G) of dimension i (0 ≤ i ≤ d) and d = Dim (RA(G)). We use this invariant in the study of groups and deduce various results. For instance, we prove the following: Theorem A.Let w be a nontrivial word in the commutator subgroup ofFn = 〈x1, …, xn〉, and letG = 〈x1, …, xn; w = 1〉. IfRSL(2, ℂ)(G)is an irreducible variety andV-1 = {ρ | ρ ∈ RSL(2, ℂ)(Fn), ρ(w) = -I} ≠ ∅, thenPd(RSL(2, ℂ)(G)) ≠ Pd(RPSL(2, ℂ)(G)). Theorem B.Let w be a nontrivial word in the free group on{x1, …, xn}with even exponent sum on each generator and exponent sum not equal to zero on at least one generator. SupposeG = 〈x1, …, xn; w = 1〉. IfRSL(2, ℂ)(G)is an irreducible variety, thenPd(RSL(2, ℂ)(G)) ≠ Pd(RPSL(2, ℂ)(G)). We also show that if G = 〈x1, . ., xn, y; W = yp〉, where p ≥ 1 and W is a word in Fn = 〈x1, …, xn〉, and A = PSL(2, ℂ), then Dim (RA(G)) = Max {3n, Dim (RA(G′)) +2 } ≤ 3n + 1 for G′ = 〈x1, …, xn; W = 1〉. Another one of our results is that if G is a torus knot group with presentation 〈x, y; xp = yt〉 then Pd(RSL(2, ℂ)(G))≠Pd(RPSL(2, ℂ)(G)).

Association of methylenetetrahydrofolate reductase genetic polymorphisms with atlantoaxial dislocation

Object Genetic mechanisms of atlantoaxial dislocation (AAD) have not previously been elucidated. The authors studied association of polymorphisms in the methylenetetrahydrofolate reductase (MTHFR) gene, which encodes enzymes of the folate pathway (implicated in causation of neural tube defects [NTDs]), in patients with AAD. Methods Molecular analysis of MTHFR polymorphisms (677C→T, cytosine to thymine and, 1298A→C, adenine to cytosine, substitutions) was carried out using polymerase chain reaction and restriction enzyme digestion in 75 consecutive patients with AAD and in their reducible (nine patients, 12%) and irreducible (66 patients, 88%) subgroups. Controls were 60 age- and sex-matched patients of the same ethnicity. Comparisons of genotype and allele frequencies were performed using a chi-square test (with significance at p < 0.05). Results The CT genotype frequency of MTHFR 677C→T polymorphism was significantly increased in the full group of patients with AAD (odds ratio [OR] 3.00, 95% confidence interval [CI] 1.28–7.14, p = 0.005) as well as in the irreducible subgroup (OR 2.81, 95% CI 1.17–6.86, p = 0.01). The frequency of T alleles was also higher in the AAD group (25.3%) than in controls (15%). The comparison of the combined frequency of CT and TT genotypes with the frequency of the CC genotype again showed significant association in AAD (OR 2.63, 95% CI 1.98–5.90, p = 0.009) and the irreducible (OR 2.5, 95% CI 1.1–5.74, p = 0.016) subgroup. There was, however, no significant association of MTHFR 1298A→C polymorphism with AAD. Conclusions Both MTHFR 677C→T polymorphism and higher T allele frequency have significant associations with AAD, especially the irreducible variety. Perhaps adequate supplementation of periconceptional folic acid to circumvent effects of this missense mutation (as is done for prevention of NTDs) would reduce the incidence of AAD.

A CLINICAL SCORING SYSTEM FOR NEUROLOGICAL ASSESSMENT OF HIGH CERVICAL MYELOPATHY

OBJECTIVE The assessment of response to treatment in pediatric patients with congenital atlantoaxial dislocation (AAD) is performed using a disability grading system but may be better determined by a score based on clinical parameters. This study proposes a scoring system based on a comprehensive neurological examination to assess surgical outcome in these patients. METHODS Sixty-seven patients with congenital AAD aged 14 years or younger were included and analyzed prospectively. A scoring system based on six factors (motor power, gait, sensory involvement, sphincteric involvement, spasticity, and respiratory difficulty) was designed at the beginning of the study and all patients were assessed using this score as well as the Di Lorenzo's grade preoperatively, postoperatively, and at the time of each follow-up visit. RESULTS There was a very high incidence of occipitalized arch of atlas and fusion of the second and third cervical vertebrae in the irreducible variety. Most patients were classified in poor grades preoperatively; however, the changes in score were seen more often when using the scoring system we developed compared with the Di Lorenzo's grade. Our score also corroborated better with the clinical improvement. CONCLUSION The clinical profiles of pediatric patients with AAD are similar with a higher incidence of atlas arch anomalies in patients with irreducible AAD. A scoring system based on clinical parameters is proposed for clinical evaluation of such patients. This system is easy to use and interpret and is more sensitive to the changes in the neurological status of patients.

The elementary theory of free pseudo p-adically closed fields of finite corank

Let be p-adic closures of a countable Hilbertian field K. The main result of [EJ] asserts that the field has the following properties for almost all σ1,…,σe + m ϵ G(K) (in the sense of the unique Haar measure on G(K)e+m):(a) Kσ is pseudo p-adically closed (abbreviation: PpC), i.e., each nonempty absolutely irreducible variety defined over Kσ has a Kσ-rational point, provided that it has a simple rational point in each p-adic closure of Kσ.(b) G(Kσ) ≅ De,m, where De,m is the free profinite product of e copies Γ1,…, Γe of G(ℚp) and a free profinite group of rank m.(c) Kσ has exactly e nonequivalent p-adic valuation rings. They are the restrictions Oσ1,…, Oσe of the unique p-adic valuation rings on , respectively.In this paper we show that this result is in a certain sense the best possible. More precisely, we first show that the class of fields which satisfy (a)–(c) above is elementary in the appropriate language e(K), which is the ordinary first-order language of rings augmented by constant symbols for the elements of K and by e new unary relation symbols (interpreted as e p-adic valuation rings).

Transfer principles for pseudo real closed e-fold ordered fields

In his famous paper [1] on the elementary theory of finite fields Ax considered fields K with the property that every absolutely irreducible variety defined over K has K-rational points. These fields have been called pseudo algebraically closed (pac) and also regularly closed, and extensively studied by Jarden, Éršov, Fried, Wheeler and others, culminating with the basic works [8] and [11].The above algebraic-geometric definition of pac fields can be put into the following equivalent model-theoretic version: K is existentially complete (ec) relative to the first order language of fields into each regular field extension of K. It has been this characterization of pac fields which the author extended in [2] to ordered fields. An ordered field (K, <) is called in [2] pseudo real closed (prc) if (K, <) is ec in every ordered field extension (L, <) with L regular over K. The concept of pre ordered field has also been introduced by McKenna in his thesis [15] by analogy with the original algebraic-geometric definition of pac fields.Given a positive integer e, a system K = (K; P1, …, Pe), where K is a field and P1, …, Pe are orders of K (identified with the corresponding positive cones), is called an e-fold ordered field (e-field). In his thesis [9] van den Dries developed a model theory for e-fields. The main result proved in [9, Chapter II] states that the theory e-OF of e-fields is model con. panionable, and the models of the model companion e-OF are explicitly described.