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This note is related to a nice short paper of M. Ochiai. We prove in a very fast way that the two-parameter family of Heegaard diagrams, constructed by Ochiai, encodes the genuine three-sphere. The result is obtained, up to isotopy, by using a sequence of only three moves in this order: a Whitehead–Zieschang reduction, a band sum and a cancellation of a handle. [ABSTRACT FROM AUTHOR]

This paper deals with Schrödinger operators as they were treated by Kato - Ikebe [3]. It is shown that every element of the domain of definition of the adjoint of such an operator has locally square integrable distributional derivatives up to the order 2. For this the potential of the Schrödinger operator must fulfil a local Stummel condition; if the potential is only locally square integrable a somewhat weaker statement is possible for three dimensions (see remark 2 at the end of this paper). [ABSTRACT FROM AUTHOR]

In this paper, we prove the boundary Lipschitz regularity and the Hopf Lemma by a unified method on Reifenberg domains for fully nonlinear elliptic equations. Precisely, if the domain Ω satisfies the exterior Reifenberg C 1 , Dini condition at x 0 ∈ ∂ Ω (see Definition 1.3), the solution is Lipschitz continuous at x 0 ; if Ω satisfies the interior Reifenberg C 1 , Dini condition at x 0 (see Definition 1.4), the Hopf lemma holds at x 0 . Our paper extends the results under the usual C 1 , Dini condition. [ABSTRACT FROM AUTHOR]

Special units are a sort of predecessor of Euler systems, and they are mainly used to obtain annihilators for class groups. So one is interested in finding as many special units as possible (actually we use a technical generalization called "semispecial"). In this paper we show that in any abelian field having a real genus field in the narrow sense all Washington units are semispecial, and that a slightly weaker statement holds true for all abelian fields. The group of Washington units is very often larger than Sinnott's group of cyclotomic units. In a companion paper we will show that in concrete families of abelian fields the group of Washington units is much larger than that of Sinnott units, by giving lower bounds on the index. Combining this with the present paper gives strong annihilation results. [ABSTRACT FROM AUTHOR]

This paper is the second of a two-part series by the author devoted to the following fundamental problem in the theory of algebraic curves in projective space: Which reducible curves arise as limits of smooth curves of general moduli? Special cases of this question studied by Sernesi (Sernesi, Edoardo (1984) On the existence of certain families of curves. Invent Math 75(1): 488 25-57), Ballico (Ballico, Edoardo (2012) Embeddings of general curves in projective spaces: the range of the quadrics. Lith Math J 52(2): 134-137), and others have been critical in the resolution of many problems in the theory of algebraic curves over the past half century. In this paper, we give sharp bounds on this problem for space curves, when the nodes are general points and the components are general in moduli. We also systematically study a variant in projective spaces of arbitrary dimension when the nodes are general in a hyperplane. The results given here significantly extend those cases established in the previous paper in this series (Eric Larson, Constructing reducible Brill-Noether curves, To appear in documentamathematica, arxiv:1603.02301), as well as those cases established by Sernesi (Sernesi, Edoardo (1984) On the existence of certain families of curves. Invent Math 75(1): 488 25-57), Ballico (Ballico, Edoardo (2012) Embeddings of general curves in projective spaces: the range of the quadrics. Lith Math J 52(2): 134-137) , and others. As explained in (Eric Larson, Degenerations of curves in projective space and the maximal rank conjecture, arXiv:1809.05980), the reducible curves constructed in this paper also play a critical role in the author's proof of the maximal rank conjecture in a subsequent paper (Eric Larson Degenerations of curves in projective space and the maximal rank conjecture, arXiv:1809.05980). [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the following singularly perturbed N-Laplacian Kirchhoff-type equation: - ε N + ε N b ∫ R N | ∇ v | N d x Δ N v + V (x) | v | N - 2 v = Q (x) f (v) , x ∈ R N , v ∈ W 1 , N (R N) , where N ≥ 2 , b is a positive constant, Δ N is the N-Laplacian operator, ε is a positive parameter, V , Q ∈ C R N , R and f has critical exponential growth at infinity. In order to overcome difficulties due to lack of compactness aroused by the critical exponential growth of f and quasilinear characteristic of the equation, we develop some delicate analyses to give a fine threshold of the Mountain-Pass minimax level and show the existence and concentration of semiclassical ground state solutions for the above equation. In particular, if V remains constant, the concentration occurs at the maximum point set of Q. [ABSTRACT FROM AUTHOR]

This paper is about one dimensional families of cyclic covers of the projective line in positive characteristic. For each such family, we study the mass formula for the number of non-ordinary curves in the family. We prove two equations for the mass formula: the first relies on tautological intersection theory; and the second relies on the a-numbers of non-ordinary curves in the family. Our results generalize the Eichler–Deuring mass formula for supersingular elliptic curves; they also generalize some theorems of Ibukiyama, Katsura, and Oort about supersingular curves of genus 2 that have an automorphism of order 3 or order 4. We determine the mass formula in many new cases, including linearized families of hyperelliptic curves of every genus and all families of cyclic covers of the projective line branched at four points. [ABSTRACT FROM AUTHOR]

The étale homotopy groups of schemes as defined by Artin and Mazur [1] have the disadvantage of being homotopy invariant only in characteristic zero. This and other related problems led to the definition of the tame topology which is coarser than the étale topology by disallowing wild ramification along the boundary of compactifications, see [9]. The objective of this paper is to show that the associated tame homotopy groups are indeed ( A 1 -)homotopy invariant, at least for regular schemes. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to present some explicit formulas for computing the Łojasiewicz exponent in the Łojasiewicz inequality comparing the rate of growth of two real bivariate analytic function germs. [ABSTRACT FROM AUTHOR]

Let C be a smooth irreducible complex projective curve of genus g ≥ 2 and M the moduli space of stable vector bundles on C of rank n and degree d with gcd (n , d) = 1 . A generalised Picard sheaf is the direct image on M of the tensor product of a universal bundle on M × C by the pullback of a vector bundle E 0 on C. In this paper, we investigate the stability of generalised Picard sheaves and, in the case where these are locally free, their deformations. When g ≥ 3 , n ≥ 2 (with some additional restrictions for g = 3 , 4 ) and the rank and degree of E 0 are coprime, this leads to the construction of a fine moduli space for deformations of Picard bundles. [ABSTRACT FROM AUTHOR]

Let X be a compact Riemann surface of genus g ≥ 2 and Ω (X) the space of holomorphic 1-forms on X. A period matrix Π is a g × g matrix defined by the pairing of a symplectic basis of H 1 (X , Z) and a basis of Ω (X) . The matrix Π is symmetric and its imaginary part Im (Π) is positive definite and depends only on symplectic bases of H 1 (X , Z) . Period matrices are important tools for studying complex structures of Riemann surfaces. However, only few formulas for period matrices are known for Riemann surfaces of genus g ≥ 2 . For Riemann surfaces of lower genus, there are only six Riemann surfaces whose period matrices are described by some formula. In this paper, our targets are algebraic curves defined by w 2 = z (z 2 - 1) (z 2 - a 1 2) (z 2 - a 2 2) ⋯ (z 2 - a g - 1 2) for any g ≥ 2 and a 1 , a 2 , ⋯ , a g - 1 ∈ R which satisfy 1 < a 1 < a 2 < ⋯ < a g - 1 . We describe period matrices of these algebraic curves by collections of complex integrals. To do this, we construct the curves from Euclidean polygons. A symplectic basis of these curves are given from the polygons. [ABSTRACT FROM AUTHOR]

The aim of the paper is to investigate the rigidity and the deformability of pseudoholomorphic curves in the nearly Kähler sphere S 6 , among minimal surfaces in spheres. Under various assumptions we describe the moduli space of all noncongruent minimal surfaces f : M → S n that are isometric to a pseudoholomorphic curve in S 6. Moreover, we prove a Schur type theorem (as reported see Chern in Curves and surfaces in Euclidean spaces. Studies in Global Geometry and Analysis, Mathematical Association of America, 1967, p. 36) for minimal surfaces in spheres. [ABSTRACT FROM AUTHOR]

In this paper we study the commensurability of hyperbolic Coxeter groups of finite covolume, providing three necessary conditions for commensurability. Moreover we provide two new sets of generators for the field of definition in case of quasi-arithmetic hyperbolic Coxeter groups. This work is a concise version of chapters 4 and 5 of the author's Ph.D. thesis Dotti (Groups of hyperbolic isometries and their commensurability. PhD thesis, Department of mathematics, University of Fribourg 2020). [ABSTRACT FROM AUTHOR]

In this paper, we consider the Schrödinger equation involving the fractional (p , p 1 , ⋯ , p m) -Laplacian as follows (- Δ) p s u + ∑ i = 1 m (- Δ) p i s u + V (ε x) (| u | (N - 2 s) / 2 s u + ∑ i = 1 m | u | p i - 2 u) = f (u) in R N , where ε is a positive parameter, N = p s , s ∈ (0 , 1) , 2 ≤ p < p 1 < ⋯ < p m < + ∞ , m ≥ 1 . The nonlinear function f has the exponential growth and potential function V is continuous function satisfying some suitable conditions. Using the penalization method and Ljusternik–Schnirelmann theory, we study the existence, multiplicity and concentration of nontrivial nonnegative solutions for small values of the parameter. In our best knowledge, it is the first time that the above problem is studied. [ABSTRACT FROM AUTHOR]

Recently there has been a lot of work on determining the Fitting ideals of arithmetic modules over groups rings, first and foremost of class groups and their Pontryagin duals. In particular, it has turned out that these Fitting ideals are usually non-principal and may be described, up to principal ideals, in terms of group-theoretical information only. The involved principal ideal factors are essentially given by values of equivariant L-functions. The present paper is not concerned with these L-functions but rather focuses on a systematic understanding of the Fitting ideals up to principal factors. To this end, we develop a certain notion of "equivalence of modules" over suitable commutative rings R. We establish that understanding the equivalence of R-modules is closely related to the classification of R-lattices. We also offer a construction of a category, inspired by derived categories, which embodies our new notion of "equivalence". [ABSTRACT FROM AUTHOR]