The Michigan Mathematical Journal Articles (Project Euclid)
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The latest articles from The Michigan Mathematical Journal on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTThu, 31 Mar 2011 11:46 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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http://projecteuclid.org/euclid.mmj/1272376024
<p><strong>Source: </strong>Michigan Math. J., Volume 59, Number 1, i--ii.</p>projecteuclid.org/euclid.mmj/1272376024_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTCorrection Terms and the Nonorientable Slice Genushttps://projecteuclid.org/euclid.mmj/1511924604<strong>Marco Golla</strong>, <strong>Marco Marengon</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 24 pages.</p><p><strong>Abstract:</strong><br/>
By considering negative surgeries on a knot $K$ in $S^{3}$ , we derive a lower bound on the nonorientable slice genus $\gamma_{4}(K)$ in terms of the signature $\sigma(K)$ and the concordance invariants $V_{i}(\overline {K})$ ; this bound strengthens a previous bound given by Batson and coincides with Ozsváth–Stipsicz–Szabó’s bound in terms of their $\upsilon$ invariant for L-space knots and quasi-alternating knots. A curious feature of our bound is superadditivity, implying, for instance, that the bound on the stable nonorientable slice genus is sometimes better than that on $\gamma_{4}(K)$ .
</p>projecteuclid.org/euclid.mmj/1511924604_20180118220312Thu, 18 Jan 2018 22:03 ESTNagata’s Compactification Theorem for Normal Toric Varieties over a Valuation Ring of Rank Onehttps://projecteuclid.org/euclid.mmj/1508983384<strong>Alejandro Soto</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 18 pages.</p><p><strong>Abstract:</strong><br/>
Using invariant Zariski–Riemann spaces, we prove that every normal toric variety over a valuation ring of rank one can be embedded as an open dense subset into a proper toric variety equivariantly. This extends a well-known theorem of Sumihiro for toric varieties over a field to this more general setting.
</p>projecteuclid.org/euclid.mmj/1508983384_20180118220312Thu, 18 Jan 2018 22:03 ESTSmooth Rational Curves on Singular Rational Surfaceshttps://projecteuclid.org/euclid.mmj/1508810820<strong>Ziquan Zhuang</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 16 pages.</p><p><strong>Abstract:</strong><br/>
We classify all complex surfaces with quotient singularities that do not contain any smooth rational curves under the assumption that the canonical divisor of the surface is not pseudo-effective. As a corollary, we show that if $X$ is a log del Pezzo surface such that, for every closed point $p\in X$ , there is a smooth curve (locally analytically) passing through $p$ , then $X$ contains at least one smooth rational curve.
</p>projecteuclid.org/euclid.mmj/1508810820_20180118220312Thu, 18 Jan 2018 22:03 ESTNielsen Realization by Gluing: Limit Groups and Free Productshttps://projecteuclid.org/euclid.mmj/1519095620<strong>Sebastian Hensel</strong>, <strong>Dawid Kielak</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 25 pages.</p><p><strong>Abstract:</strong><br/> We generalize the Karrass–Pietrowski–Solitar and the Nielsen realization theorems from the setting of free groups to that of free products. As a result, we obtain a fixed point theorem for finite groups of outer automorphisms acting on the relative free splitting complex of Handel and Mosher and on the outer space of a free product of Guirardel and Levitt, and also a relative version of the Nielsen realization theorem, which, in the case of free groups, answers a question of Karen Vogtmann. We also prove Nielsen realization for limit groups and, as a byproduct, obtain a new proof that limit groups are CAT( $0$ ). The proofs rely on a new version of Stallings’ theorem on groups with at least two ends, in which some control over the behavior of virtual free factors is gained. </p>projecteuclid.org/euclid.mmj/1519095620_20180219220038Mon, 19 Feb 2018 22:00 ESTCharacterization of Aleksandrov Spaces of Curvature Bounded Above by Means of the Metric Cauchy–Schwarz Inequalityhttps://projecteuclid.org/euclid.mmj/1519095621<strong>I. D. Berg</strong>, <strong>Igor G. Nikolaev</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 44 pages.</p><p><strong>Abstract:</strong><br/>
We consider the previously introduced notion of the $K$ -quadrilateral cosine, which is the cosine under parallel transport in model $K$ -space, and which is denoted by $\operatorname{cosq}_{K}$ . In $K$ -space, $\vert \operatorname{cosq}_{K}\vert \leq 1$ is equivalent to the Cauchy–Schwarz inequality for tangent vectors under parallel transport. Our principal result states that a geodesically connected metric space (of diameter not greater than $\pi /(2\sqrt{K})$ if $K\gt 0$ ) is an $\Re_{K}$ domain (otherwise known as a $\operatorname{CAT}(K)$ space) if and only if always $\operatorname{cosq}_{K}\leq 1$ or always $\operatorname{cosq}_{K}\geq -1$ . (We prove that in such spaces always $\operatorname{cosq}_{K}\leq 1$ is equivalent to always $\operatorname{cosq}_{K}\geq -1$ .) The case of $K=0$ was treated in our previous paper on quasilinearization. We show that in our theorem the diameter hypothesis for positive $K$ is sharp, and we prove an extremal theorem—isometry with a section of $K$ -plane—when $\vert \operatorname{cosq}_{K}\vert $ attains an upper bound of $1$ , the case of equality in the metric Cauchy–Schwarz inequality. We derive from our main theorem and our previous result for $K=0$ a complete solution of Gromov’s curvature problem in the context of Aleksandrov spaces of curvature bounded above.
</p>projecteuclid.org/euclid.mmj/1519095621_20180219220038Mon, 19 Feb 2018 22:00 ESTA Note on Brill–Noether Existence for Graphs of Low Genushttps://projecteuclid.org/euclid.mmj/1519095622<strong>Stanislav Atanasov</strong>, <strong>Dhruv Ranganathan</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 24 pages.</p><p><strong>Abstract:</strong><br/>
In an influential 2008 paper, Baker proposed a number of conjectures relating the Brill–Noether theory of algebraic curves with a divisor theory on finite graphs. In this note, we examine Baker’s Brill–Noether existence conjecture for special divisors. For $g\leq5$ and $\rho(g,r,d)$ nonnegative, every graph of genus $g$ is shown to admit a divisor of rank $r$ and degree at most $d$ . As further evidence, the conjecture is shown to hold in rank $1$ for a number families of highly connected combinatorial types of graphs. In the relevant genera, our arguments give the first combinatorial proof of the Brill–Noether existence theorem for metric graphs, giving a partial answer to a related question of Baker.
</p>projecteuclid.org/euclid.mmj/1519095622_20180219220038Mon, 19 Feb 2018 22:00 ESTSplitting Criteria for Vector Bundles Induced by Restrictions to Divisorshttps://projecteuclid.org/euclid.mmj/1521856929<strong>Mihai Halic</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 67, Number 2, 227--251.</p><p><strong>Abstract:</strong><br/>
In this article we obtain criteria for the splitting and triviality of vector bundles by restricting them to partially ample divisors. This allows us to study the problem of splitting on the total space of fibre bundles. The statements are illustrated with examples.
For products of minuscule homogeneous varieties, we show that the splitting of vector bundles can be tested by restricting them to subproducts of Schubert $2$ -planes. By using known cohomological criteria for multiprojective spaces, we deduce necessary and sufficient conditions for the splitting of vector bundles on products of minuscule varieties.
The triviality criteria are particularly suited to Frobenius split varieties. We prove that a vector bundle on a smooth toric variety, whose anticanonical bundle has stable base locus of codimension at least three, is trivial precisely when its restrictions to the invariant divisors are trivial, with trivializations compatible along the various intersections.
</p>projecteuclid.org/euclid.mmj/1521856929_20180511220024Fri, 11 May 2018 22:00 EDTGivental-Type Reconstruction at a Nonsemisimple Pointhttps://projecteuclid.org/euclid.mmj/1523584849<strong>Alexey Basalaev</strong>, <strong>Nathan Priddis</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 67, Number 2, 333--369.</p><p><strong>Abstract:</strong><br/>
We consider the orbifold curve that is a quotient of an elliptic curve $\mathcal{E}$ by a cyclic group of order 4. We develop a systematic way to obtain a Givental-type reconstruction of Gromov–Witten theory of the orbifold curve via the product of the Gromov–Witten theories of a point. This is done by employing mirror symmetry and certain results in FJRW theory. In particular, we present the particular Givental’s action giving the CY/LG correspondence between the Gromov–Witten theory of the orbifold curve $\mathcal{E}/\mathbb{Z}_{4}$ and FJRW theory of the pair defined by the polynomial $x^{4}+y^{4}+z^{2}$ and the maximal group of diagonal symmetries. The methods we have developed can easily be applied to other finite quotients of an elliptic curve. Using Givental’s action, we also recover this FJRW theory via the product of the Gromov–Witten theories of a point. Combined with the CY/LG action, we get a result in “pure” Gromov–Witten theory with the help of modern mirror symmetry conjectures.
</p>projecteuclid.org/euclid.mmj/1523584849_20180511220024Fri, 11 May 2018 22:00 EDTDiophantine Approximation Constants for Varieties over Function Fieldshttps://projecteuclid.org/euclid.mmj/1522980164<strong>Nathan Grieve</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 67, Number 2, 371--404.</p><p><strong>Abstract:</strong><br/>
By analogy with the program of McKinnon and Roth [10], we define and study approximation constants for points of a projective variety $X$ defined over $\mathbf {K}$ , the function field of an irreducible and nonsingular in codimension $1$ projective variety defined over an algebraically closed field of characteristic zero. In this setting, we use Wang’s theorem, which is an effective version of Schmidt’s subspace theorem, to give a sufficient condition for such approximation constants to be computed on a proper $\mathbf{K}$ -subvariety of $X$ . We also indicate how our approximation constants are related to volume functions and Seshadri constants.
</p>projecteuclid.org/euclid.mmj/1522980164_20180511220024Fri, 11 May 2018 22:00 EDTLegendrian Lens Space Surgerieshttps://projecteuclid.org/euclid.mmj/1522980162<strong>Hansjörg Geiges</strong>, <strong>Sinem Onaran</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 67, Number 2, 405--422.</p><p><strong>Abstract:</strong><br/>
We show that every tight contact structure on any of the lens spaces $L(ns^{2}-s+1,s^{2})$ with $n\geq 2$ and $s\geq 1$ can be obtained by a single Legendrian surgery along a suitable Legendrian realisation of the negative torus knot $T(s,-(sn-1))$ in the tight or an overtwisted contact structure on the $3$ -sphere.
</p>projecteuclid.org/euclid.mmj/1522980162_20180511220024Fri, 11 May 2018 22:00 EDTOn the $\operatorname{Pin}(2)$ -Equivariant Monopole Floer Homology of Plumbed 3-Manifoldshttps://projecteuclid.org/euclid.mmj/1523498585<strong>Irving Dai</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 67, Number 2, 423--447.</p><p><strong>Abstract:</strong><br/>
We compute the $\operatorname{Pin}(2)$ -equivariant monopole Floer homology for the class of plumbed 3-manifolds considered by Ozsváth and Szabó [18]. We show that for these manifolds, the $\operatorname{Pin}(2)$ -equivariant monopole Floer homology can be calculated in terms of the Heegaard Floer/monopole Floer lattice complex defined by Némethi [15]. Moreover, we prove that in such cases the ranks of the usual monopole Floer homology groups suffice to determine both the Manolescu correction terms and the $\operatorname{Pin}(2)$ -homology as an Abelian group. As an application, we show that $\beta(-Y,s)=\bar{\mu}(Y,s)$ for all plumbed 3-manifolds with at most one “bad” vertex, proving (an analogue of) a conjecture posed by Manolescu [12]. Our proof also generalizes results by Stipsicz [21] and Ue [26] relating $\bar{\mu}$ with the Ozsváth–Szabó $d$ -invariant. Some observations aimed at extending our computations to manifolds with more than one bad vertex are included at the end of the paper.
</p>projecteuclid.org/euclid.mmj/1523498585_20180511220024Fri, 11 May 2018 22:00 EDTSimultaneous Flips on Triangulated Surfaceshttps://projecteuclid.org/euclid.mmj/1530151253<strong>Valentina Disarlo</strong>, <strong>Hugo Parlier</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 67, Number 3, 451--464.</p><p><strong>Abstract:</strong><br/>
We investigate a type of distance between triangulations on finite-type surfaces where one moves between triangulations by performing simultaneous flips. We consider triangulations up to homeomorphism, and our main results are upper bounds on the distance between triangulations that only depend on the topology of the surface.
</p>projecteuclid.org/euclid.mmj/1530151253_20180801220102Wed, 01 Aug 2018 22:01 EDTMirror Theorem for Elliptic Quasimap Invariants of Local Calabi–Yau Varietieshttps://projecteuclid.org/euclid.mmj/1529114457<strong>Hyenho Lho</strong>, <strong>Jeongseok Oh</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 67, Number 3, 465--484.</p><p><strong>Abstract:</strong><br/>
The elliptic quasi-map potential function is explicitly calculated for Calabi–Yau complete intersections in projective spaces in [13]. We extend this result to local Calabi–Yau varieties. Using this and the wall crossing formula in [5], we can calculate the elliptic Gromov–Witten potential function.
</p>projecteuclid.org/euclid.mmj/1529114457_20180801220102Wed, 01 Aug 2018 22:01 EDTA Geometric Reverse to the Plus Construction and Some Examples of Pseudocollars on High-Dimensional Manifoldshttps://projecteuclid.org/euclid.mmj/1522980163<strong>Jeffrey J. Rolland</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 67, Number 3, 485--509.</p><p><strong>Abstract:</strong><br/>
In this paper, we develop a geometric procedure for producing a “reverse” to Quillen’s plus construction, a construction called a $1$ -sided $h$ -cobordism or semi- $h$ -cobordism . We then use this reverse to the plus construction to produce uncountably many distinct ends of manifolds called pseudocollars , which are stackings of $1$ -sided $h$ -cobordisms. Each of our pseudocollars has the same boundary and prohomology systems at infinity and similar group-theoretic properties for their profundamental group systems at infinity. In particular, the kernel group of each group extension for each $1$ -sided $h$ -cobordism in the pseudocollars is the same group. Nevertheless, the profundamental group systems at infinity are all distinct. A good deal of combinatorial group theory is needed to verify this fact, including an application of Thompson’s group $V$ .
The notion of pseudocollars originated in Hilbert cube manifold theory, where it was part of a necessary and sufficient condition for placing a $\mathcal{Z}$ -set as the boundary of an open Hilbert cube manifold.
</p>projecteuclid.org/euclid.mmj/1522980163_20180801220102Wed, 01 Aug 2018 22:01 EDTConnected Components of the Moduli of Elliptic $K3$ Surfaceshttps://projecteuclid.org/euclid.mmj/1528941621<strong>Ichiro Shimada</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 67, Number 3, 511--559.</p><p><strong>Abstract:</strong><br/>
The combinatorial type of an elliptic $K3$ surface with a zero section is the pair of the $\mathit{ADE}$ -type of the singular fibers and the torsion part of the Mordell–Weil group. We determine the set of connected components of the moduli of elliptic $K3$ surfaces with fixed combinatorial type. Our method relies on the theory of Miranda and Morrison on the structure of a genus of even indefinite lattices and on computer-aided calculations of $p$ -adic quadratic forms.
</p>projecteuclid.org/euclid.mmj/1528941621_20180801220102Wed, 01 Aug 2018 22:01 EDTExplicit Björling Surfaces with Prescribed Geometryhttps://projecteuclid.org/euclid.mmj/1531447375<strong>Rafael López</strong>, <strong>Matthias Weber</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 67, Number 3, 561--584.</p><p><strong>Abstract:</strong><br/>
We develop a new method to construct explicit regular minimal surfaces in Euclidean space that are defined on the entire complex plane with controlled geometry. More precisely, we show that for a large class of planar curves $(x(t),y(t))$ , we can find a third coordinate $z(t)$ and normal fields $n(t)$ along the space curve $c(t)=(x(t),y(t),z(t))$ so that the Björling formula applied to $c(t)$ and $n(t)$ can be explicitly evaluated. We give many examples.
</p>projecteuclid.org/euclid.mmj/1531447375_20180801220102Wed, 01 Aug 2018 22:01 EDTManifolds Which Admit Maps with Finitely Many Critical Points Into Spheres of Small Dimensionshttps://projecteuclid.org/euclid.mmj/1529460326<strong>Louis Funar</strong>, <strong>Cornel Pintea</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 67, Number 3, 585--615.</p><p><strong>Abstract:</strong><br/>
We construct, for $m\geq6$ and $2n\leq m$ , closed manifolds $M^{m}$ with finite nonzero $\varphi(M^{m},S^{n}$ ), where $\varphi(M,N)$ denotes the minimum number of critical points of a smooth map $M\to N$ . We also give some explicit families of examples for even $m\geq6$ and $n=3$ , taking advantage of the Lie group structure on $S^{3}$ . Moreover, there are infinitely many such examples with $\varphi(M^{m},S^{n})=1$ . Eventually, we compute the signature of the manifolds $M^{2n}$ occurring for even $n$ .
</p>projecteuclid.org/euclid.mmj/1529460326_20180801220102Wed, 01 Aug 2018 22:01 EDTTree-Lattice Zeta Functions and Class Numbershttps://projecteuclid.org/euclid.mmj/1529460323<strong>Anton Deitmar</strong>, <strong>Ming-Hsuan Kang</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 67, Number 3, 617--645.</p><p><strong>Abstract:</strong><br/>
We extend the theory of Ihara zeta functions to noncompact arithmetic quotients of Bruhat–Tits trees. This new zeta function turns out to be a rational function despite the infinite-dimensional setting. In general, it has zeros and poles in contrast to the compact case. The determinant formulas of Bass and Ihara hold if we define the determinant as the limit of all finite principal minors. From this analysis we derive a prime geodesic theorem, which, applied to special arithmetic groups, yields new asymptotic assertions on class numbers of orders in global fields.
</p>projecteuclid.org/euclid.mmj/1529460323_20180801220102Wed, 01 Aug 2018 22:01 EDTExtensions of Some Classical Local Moves on Knot Diagramshttps://projecteuclid.org/euclid.mmj/1531447373<strong>Benjamin Audoux</strong>, <strong>Paolo Bellingeri</strong>, <strong>Jean-Baptiste Meilhan</strong>, <strong>Emmanuel Wagner</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 67, Number 3, 647--672.</p><p><strong>Abstract:</strong><br/>
We consider local moves on classical and welded diagrams: (self-)crossing change, (self-)virtualization, virtual conjugation, Delta, fused, band-pass, and welded band-pass moves. Interrelationships between these moves are discussed, and, for each of these moves, we provide an algebraic classification. We address the question of relevant welded extensions for classical moves in the sense that the classical quotient of classical object embeds into the welded quotient of welded objects. As a byproduct, we obtain that all of the local moves mentioned are unknotting operations for welded (long) knots. We also mention some topological interpretations for these combinatorial quotients.
</p>projecteuclid.org/euclid.mmj/1531447373_20180801220102Wed, 01 Aug 2018 22:01 EDTGromov–Witten Invariants of the Hilbert Scheme of Two Points on a Hirzebruch Surfacehttps://projecteuclid.org/euclid.mmj/1538532103<strong>Yong Fu</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 67, Number 4, 675--713.</p><p><strong>Abstract:</strong><br/>
In Gromov–Witten theory the virtual localization method is used only when the invariant curves are isolated under a torus action. In this paper, we explore a strategy to apply the localization formula to compute the Gromov–Witten invariants by carefully choosing the related cycles to circumvent the continuous families of invariant curves when there are any. For the example of the two-pointed Hilbert scheme of Hirzebruch surface $F_{1}$ , we manage to compute some Gromov–Witten invariants, and then by combining with the associativity law of (small) quantum cohomology ring, we succeed in computing all 1- and 2-pointed Gromov–Witten invariants of genus 0 of the Hilbert scheme with the help of [13].
</p>projecteuclid.org/euclid.mmj/1538532103_20181115220444Thu, 15 Nov 2018 22:04 ESTCounting the Ideals of Given Codimension of the Algebra of Laurent Polynomials in Two Variableshttps://projecteuclid.org/euclid.mmj/1529114453<strong>Christian Kassel</strong>, <strong>Christophe Reutenauer</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 67, Number 4, 715--741.</p><p><strong>Abstract:</strong><br/>
We establish an explicit formula for the number $C_{n}(q)$ of ideals of codimension (colength) $n$ of the algebra $\mathbb{F}_{q}[x,y,x^{-1},y^{-1}]$ of Laurent polynomials in two variables over a finite field $\mathbb{F}_{q}$ of cardinality $q$ . This number is a palindromic polynomial of degree $2n$ in $q$ . Moreover, $C_{n}(q)=(q-1)^{2}P_{n}(q)$ , where $P_{n}(q)$ is another palindromic polynomial; the latter is a $q$ -analogue of the sum of divisors of $n$ , which happens to be the number of subgroups of $\mathbb{Z}^{2}$ of index $n$ .
</p>projecteuclid.org/euclid.mmj/1529114453_20181115220444Thu, 15 Nov 2018 22:04 ESTExtactic Divisors for Webs and Lines on Projective Surfaceshttps://projecteuclid.org/euclid.mmj/1531447376<strong>Maycol Falla Luza</strong>, <strong>Jorge Vitório Pereira</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 67, Number 4, 743--756.</p><p><strong>Abstract:</strong><br/>
Given a web (multifoliation) and a linear system on a projective surface, we construct divisors cutting out the locus where some element of the linear system has abnormal contact with the leaf of the web. We apply these ideas to reobtain a classical result by Salmon on the number of lines on a projective surface. In a different vein, we investigate the numbers of lines and disjoint lines contained in a projective surface and tangent to a contact distribution.
</p>projecteuclid.org/euclid.mmj/1531447376_20181115220444Thu, 15 Nov 2018 22:04 ESTDyadic Representation and Boundedness of Nonhomogeneous Calderón–Zygmund Operators with Mild Kernel Regularityhttps://projecteuclid.org/euclid.mmj/1531447374<strong>Ana Grau de la Herrán</strong>, <strong>Tuomas Hytönen</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 67, Number 4, 757--786.</p><p><strong>Abstract:</strong><br/>
We prove a new dyadic representation theorem with applications to the $T(1)$ and $A_{2}$ theorems. In particular, we obtain the nonhomogeneous $T(1)$ theorem under weaker kernel regularity than in the earlier approaches.
</p>projecteuclid.org/euclid.mmj/1531447374_20181115220444Thu, 15 Nov 2018 22:04 ESTFamilies of Elliptic Curves in $\mathbb{P}^{3}$ and Bridgeland Stabilityhttps://projecteuclid.org/euclid.mmj/1538705132<strong>Patricio Gallardo</strong>, <strong>César Lozano Huerta</strong>, <strong>Benjamin Schmidt</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 67, Number 4, 787--813.</p><p><strong>Abstract:</strong><br/>
We study wall crossings in Bridgeland stability for the Hilbert scheme of elliptic quartic curves in the three-dimensional projective space. We provide a geometric description of each of the moduli spaces we encounter, including when the second component of this Hilbert scheme appears. Along the way, we prove that the principal component of this Hilbert scheme is a double blowup with smooth centers of a Grassmannian, exhibiting a completely different proof of this known result by Avritzer and Vainsencher. This description allows us to compute the cone of effective divisors of this component.
</p>projecteuclid.org/euclid.mmj/1538705132_20181115220444Thu, 15 Nov 2018 22:04 ESTHenkin Measures for the Drury–Arveson Spacehttps://projecteuclid.org/euclid.mmj/1539072025<strong>Michael Hartz</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 67, Number 4, 815--826.</p><p><strong>Abstract:</strong><br/>
We exhibit Borel probability measures on the unit sphere in ${\mathbb{C}}^{d}$ for $d\ge2$ that are Henkin for the multiplier algebra of the Drury–Arveson space, but not Henkin in the classical sense. This provides a negative answer to a conjecture of Clouâtre and Davidson.
</p>projecteuclid.org/euclid.mmj/1539072025_20181115220444Thu, 15 Nov 2018 22:04 ESTRelative $\mathbb{Q}$ -Gradings from Bordered Floer Theoryhttps://projecteuclid.org/euclid.mmj/1539763499<strong>Robert Lipshitz</strong>, <strong>Peter Ozsváth</strong>, <strong>Dylan P. Thurston</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 67, Number 4, 827--838.</p><p><strong>Abstract:</strong><br/>
In this paper, we show how to recover the relative $\mathbb{Q}$ -grading in Heegaard Floer homology from the noncommutative grading on bordered Floer homology.
</p>projecteuclid.org/euclid.mmj/1539763499_20181115220444Thu, 15 Nov 2018 22:04 ESTEffective Divisors in $\overline{\mathcal{M}}_{g,n}$ from Abelian Differentialshttps://projecteuclid.org/euclid.mmj/1542337465<strong>Scott Mullane</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 67, Number 4, 839--889.</p><p><strong>Abstract:</strong><br/>
We compute many new classes of effective divisors in $\overline {\mathcal {M}}_{g,n}$ coming from the strata of Abelian differentials. Our method utilizes maps between moduli spaces and the degeneration of Abelian differentials.
</p>projecteuclid.org/euclid.mmj/1542337465_20181115220444Thu, 15 Nov 2018 22:04 ESTIndexhttps://projecteuclid.org/euclid.mmj/1542337466<p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 67, Number 4, 893--895.</p>projecteuclid.org/euclid.mmj/1542337466_20181115220444Thu, 15 Nov 2018 22:04 ESTThe Homotopy Lie Algebra of Symplectomorphism Groups of 3-Fold Blowups of $(S^{2}\times S^{2},\sigma_{\mathrm{std}}\oplus\sigma_{\mathrm{std}})$https://projecteuclid.org/euclid.mmj/1547089467<strong>Sílvia Anjos</strong>, <strong>Sinan Eden</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 56 pages.</p><p><strong>Abstract:</strong><br/>
We consider the 3-point blowup of the manifold $(S^{2}\times S^{2},\sigma\oplus\sigma)$ , where $\sigma$ is the standard symplectic form that gives area 1 to the sphere $S^{2}$ , and study its group of symplectomorphisms $\operatorname {Symp}(S^{2}\times S^{2}\#3\overline{\mathbb{C}\mathbb{P}}^{2},\omega)$ . So far, the monotone case was studied by Evans [6], who proved that this group is contractible. Moreover, Li, Li, and Wu [13] showed that the group $\operatorname {Symp}_{h}(S^{2}\times S^{2}\#3\overline{\mathbb{C}\mathbb{P}}^{2},\omega)$ of symplectomorphisms that act trivially on homology is always connected, and recently, in [14], they also computed its fundamental group. We describe, in full detail, the rational homotopy Lie algebra of this group.
We show that some particular circle actions contained in the symplectomorphism group generate its full topology. More precisely, they give the generators of the homotopy graded Lie algebra of $\operatorname {Symp}(S^{2}\times S^{2}\#3\overline{\mathbb{C}\mathbb{P}}^{2},\omega)$ . Our study depends on Karshon’s classification of Hamiltonian circle actions and the inflation technique introduced by Lalonde and McDuff. As an application, we deduce the rank of the homotopy groups of $\operatorname {Symp}(\mathbb{C}\mathbb{P}^{2}\#5\overline{\mathbb{C}\mathbb{P}}^{2},\widetilde{\omega})$ in the case of small blowups.
</p>projecteuclid.org/euclid.mmj/1547089467_20190109220447Wed, 09 Jan 2019 22:04 ESTBiharmonic Maps on Principal $G$ -Bundles over Complete Riemannian Manifolds of Nonpostive Ricci Curvaturehttps://projecteuclid.org/euclid.mmj/1547089468<strong>Hajime Urakawa</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 13 pages.</p><p><strong>Abstract:</strong><br/>
We show that, for every principal $G$ -bundle over a complete Riemannian manifold of nonpositive Ricci curvature, if the projection of the $G$ -bundle is biharmonic, then it is harmonic.
</p>projecteuclid.org/euclid.mmj/1547089468_20190109220447Wed, 09 Jan 2019 22:04 ESTAlmost Abelian Artin Representations of $\mathbb{Q}$https://projecteuclid.org/euclid.mmj/1545037225<strong>David E. Rohrlich</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 19 pages.</p>projecteuclid.org/euclid.mmj/1545037225_20190109220447Wed, 09 Jan 2019 22:04 ESTOn a Conjecture of Sokal Concerning Roots of the Independence Polynomialhttps://projecteuclid.org/euclid.mmj/1541667626<strong>Han Peters</strong>, <strong>Guus Regts</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 23 pages.</p><p><strong>Abstract:</strong><br/>
A conjecture of Sokal [24], regarding the domain of nonvanishing for independence polynomials of graphs, states that given any natural number $\Delta\ge3$ , there exists a neighborhood in $\mathbb{C}$ of the interval $[0,(\Delta-1)^{\Delta-1}/(\Delta-2)^{\Delta})$ on which the independence polynomial of any graph with maximum degree at most $\Delta$ does not vanish. We show here that Sokal’s conjecture holds, as well as a multivariate version, and prove the optimality for the domain of nonvanishing. An important step is to translate the setting to the language of complex dynamical systems.
</p>projecteuclid.org/euclid.mmj/1541667626_20190109220447Wed, 09 Jan 2019 22:04 ESTEtemadi and Kolmogorov Inequalities in Noncommutative Probability Spaceshttps://projecteuclid.org/euclid.mmj/1541667627<strong>Ali Talebi</strong>, <strong>Mohammad Sal Moslehian</strong>, <strong>Ghadir Sadeghi</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 13 pages.</p><p><strong>Abstract:</strong><br/>
Based on a maximal inequality-type result of Cuculescu, we establish some noncommutative maximal inequalities such as the Hajék–Penyi and Etemadi inequalities. In addition, we present a noncommutative Kolmogorov-type inequality by showing that if $x_{1},x_{2},\ldots,x_{n}$ are successively independent self-adjoint random variables in a noncommutative probability space $(\mathfrak{M},\tau)$ such that $\tau(x_{k})=0$ and $s_{k}s_{k-1}=s_{k-1}s_{k}$ , where $s_{k}=\sum_{j=1}^{k}x_{j}$ , then, for any $\lambda\gt 0$ , there exists a projection $e$ such that
\[1-\frac{(\lambda+\max_{1\leq k\leq n}\Vert x_{k}\Vert )^{2}}{\sum_{k=1}^{n}\operatorname{var}(x_{k})}\leq\tau(e)\leq\frac{\tau(s_{n}^{2})}{\lambda^{2}}.\] As a result, we investigate the relation between the convergence of a series of independent random variables and the corresponding series of their variances.
</p>projecteuclid.org/euclid.mmj/1541667627_20190109220447Wed, 09 Jan 2019 22:04 ESTOn Almost Everywhere Convergence of Tensor Product Spline Projectionshttps://projecteuclid.org/euclid.mmj/1541667630<strong>Markus Passenbrunner</strong>, <strong>Joscha Prochno</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 15 pages.</p><p><strong>Abstract:</strong><br/>
Let $d\in \mathbb{N}$ , and let $f$ be a function in the Orlicz class $L(\log^{+}L)^{d-1}$ defined on the unit cube $[0,1]^{d}$ in $\mathbb{R}^{d}$ . Given knot sequences $\Delta_{1},\ldots,\Delta_{d}$ on $[0,1]$ , we first prove that the orthogonal projection $P_{(\Delta_{1},\ldots,\Delta_{d})}(f)$ onto the space of tensor product splines with arbitrary orders $(k_{1},\ldots,k_{d})$ and knots $\Delta_{1},\ldots,\Delta_{d}$ converges to $f$ almost everywhere as the mesh diameters $|\Delta_{1}|,\ldots,|\Delta_{d}|$ tend to zero. This extends the one-dimensional result in [9] to arbitrary dimensions.
In the second step, we show that this result is optimal, that is, given any “bigger” Orlicz class $X=\sigma (L)L(\log^{+}L)^{d-1}$ with an arbitrary function $\sigma $ tending to zero at infinity, there exist a function $\varphi \in X$ and partitions of the unit cube such that the orthogonal projections of $\varphi $ do not converge almost everywhere.
</p>projecteuclid.org/euclid.mmj/1541667630_20190109220447Wed, 09 Jan 2019 22:04 ESTNote on MacPherson’s Local Euler Obstructionhttps://projecteuclid.org/euclid.mmj/1548817530<strong>Yunfeng Jiang</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 24 pages.</p><p><strong>Abstract:</strong><br/>
This is a note on MacPherson’s local Euler obstruction, which plays an important role recently in the Donaldson–Thomas theory by the work of Behrend.
We introduce MacPherson’s original definition and prove that it is equivalent to the algebraic definition used by Behrend, following the method of González-Sprinberg. We also give a formula of the local Euler obstruction in terms of Lagrangian intersections. As an application, we consider a scheme or DM stack $X$ admitting a symmetric obstruction theory. Furthermore, we assume that there is a $\mathbb{C}^{*}$ action on $X$ that makes the obstruction theory $\mathbb{C}^{*}$ -equivariant. The $\mathbb{C}^{*}$ -action on the obstruction theory naturally gives rise to a cosection map in the Kiem–Li sense. We prove that Behrend’s weighted Euler characteristic of $X$ is the same as the Kiem–Li localized invariant of $X$ by the $\mathbb{C}^{*}$ -action.
</p>projecteuclid.org/euclid.mmj/1548817530_20190129220559Tue, 29 Jan 2019 22:05 ESTLattice Simplices of Maximal Dimension with a Given Degreehttps://projecteuclid.org/euclid.mmj/1548817531<strong>Akihiro Higashitani</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 18 pages.</p><p><strong>Abstract:</strong><br/>
It was proved by Nill that for any lattice simplex of dimension $d$ with degree $s$ that is not a lattice pyramid, we have $d+1\leq 4s-1$ . In this paper, we give a complete characterization of lattice simplices satisfying the equality, that is, the lattice simplices of dimension $4s-2$ with degree $s$ that are not lattice pyramids. It turns out that such simplices arise from binary simplex codes. As an application of this characterization, we show that such simplices are counterexamples for the conjecture known as the Cayley conjecture. Moreover, by slightly modifying Nill’s inequality we also see the sharper bound $d+1\leq f(2s)$ , where $f(M)=\sum_{n=0}^{\lfloor \log_{2}M\rfloor }\lfloor M/2^{n}\rfloor $ for $M\in \mathbb{Z}_{\geq 0}$ . We also observe that any lattice simplex attaining this sharper bound always comes from a binary code.
</p>projecteuclid.org/euclid.mmj/1548817531_20190129220559Tue, 29 Jan 2019 22:05 ESTStrong Factorizations of Operators with Applications to Fourier and Cesàro Transformshttps://projecteuclid.org/euclid.mmj/1548817532<strong>O. Delgado</strong>, <strong>M. Mastyło</strong>, <strong>E. A. Sánchez Pérez</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 26 pages.</p><p><strong>Abstract:</strong><br/>
Consider two continuous linear operators $T\colon X_{1}(\mu )\to Y_{1}(\nu )$ and $S\colon X_{2}(\mu )\to Y_{2}(\nu )$ between Banach function spaces related to different $\sigma $ -finite measures $\mu $ and $\nu $ . By means of weighted norm inequalities we characterize when $T$ can be strongly factored through $S$ , that is, when there exist functions $g$ and $h$ such that $T(f)=gS(hf)$ for all $f\in X_{1}(\mu )$ . For the case of spaces with Schauder basis, our characterization can be improved, as we show when $S$ is, for instance, the Fourier or Cesàro operator. Our aim is to study the case where the map $T$ is besides injective. Then we say that it is a representing operator—in the sense that it allows us to represent each element of the Banach function space $X(\mu )$ by a sequence of generalized Fourier coefficients—providing a complete characterization of these maps in terms of weighted norm inequalities. We also provide some examples and applications involving recent results on the Hausdorff–Young and the Hardy–Littlewood inequalities for operators on weighted Banach function spaces.
</p>projecteuclid.org/euclid.mmj/1548817532_20190129220559Tue, 29 Jan 2019 22:05 ESTOn Diamond’s $L^{1}$ Criterion for Asymptotic Density of Beurling Generalized Integershttps://projecteuclid.org/euclid.mmj/1548903624<strong>Gregory Debruyne</strong>, <strong>Jasson Vindas</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 13 pages.</p><p><strong>Abstract:</strong><br/>
We give a short proof of the $L^{1}$ criterion for Beurling generalized integers to have a positive asymptotic density. We in fact prove the existence of density under a weaker hypothesis. We also discuss related sufficient conditions for the estimate $m(x)=\sum_{n_{k}\leq x}\mu (n_{k})/n_{k}=o(1)$ with the Beurling analog $\mu $ of the Möbius function.
</p>projecteuclid.org/euclid.mmj/1548903624_20190130220051Wed, 30 Jan 2019 22:00 ESTClassification of Systems with Center-Stable Torihttps://projecteuclid.org/euclid.mmj/1549681298<strong>Andy Hammerlindl</strong>, <strong>Rafael Potrie</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 20 pages.</p><p><strong>Abstract:</strong><br/>
We give a classification of three-dimensional partially hyperbolic systems that have at least one torus tangent to the center-stable bundle.
</p>projecteuclid.org/euclid.mmj/1549681298_20190208220214Fri, 08 Feb 2019 22:02 ESTExtremal Rays and Nefness of Tangent Bundleshttps://projecteuclid.org/euclid.mmj/1549681299<strong>Akihiro Kanemitsu</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 22 pages.</p><p><strong>Abstract:</strong><br/>
In view of Mori theory, rational homogenous manifolds satisfy a recursive condition: every elementary contraction is a rational homogeneous fibration, and the image of any elementary contraction also satisfies the same property. In this paper, we show that a smooth Fano $n$ -fold with the same condition and Picard number greater than $n-6$ is either a rational homogeneous manifold or the product of $n-7$ copies of $\mathbb{P}^{1}$ and a Fano $7$ -fold $X_{0}$ constructed by G. Ottaviani. We also clarify that $X_{0}$ has a non-nef tangent bundle and in particular is not rational homogeneous.
</p>projecteuclid.org/euclid.mmj/1549681299_20190208220214Fri, 08 Feb 2019 22:02 ESTFat Flats in Rank One Manifoldshttps://projecteuclid.org/euclid.mmj/1549681300<strong>D. Constantine</strong>, <strong>J.-F. Lafont</strong>, <strong>D. B. McReynolds</strong>, <strong>D. J. Thompson</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 25 pages.</p><p><strong>Abstract:</strong><br/>
We study closed nonpositively curved Riemannian manifolds $M$ that admit “fat $k$ -flats”; that is, the universal cover $\tilde{M}$ contains a positive-radius neighborhood of a $k$ -flat on which the sectional curvatures are identically zero. We investigate how the fat $k$ -flats affect the cardinality of the collection of closed geodesics. Our first main result is to construct rank $1$ nonpositively curved manifolds with a fat $1$ -flat that corresponds to a twisted cylindrical neighborhood of a geodesic on $M$ . As a result, $M$ contains an embedded closed geodesic with a flat neighborhood, but $M$ nevertheless has only countably many closed geodesics. Such metrics can be constructed on finite covers of arbitrary odd-dimensional finite volume hyperbolic manifolds. Our second main result is a proof of a closing theorem for fat flats, which implies that a manifold $M$ with a fat $k$ -flat contains an immersed, totally geodesic $k$ -dimensional flat closed submanifold. This guarantees the existence of uncountably many closed geodesics when $k\geq 2$ . Finally, we collect results on thermodynamic formalism for the class of manifolds considered in this paper.
</p>projecteuclid.org/euclid.mmj/1549681300_20190208220214Fri, 08 Feb 2019 22:02 ESTA Uniform Bound on the Brauer Groups of Certain log K3 Surfaceshttps://projecteuclid.org/euclid.mmj/1550480562<strong>Martin Bright</strong>, <strong>Julian Lyczak</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 8 pages.</p><p><strong>Abstract:</strong><br/>
Let $U$ be the complement of a smooth anticanonical divisor in a del Pezzo surface of degree at most 7 over a number field $k$ . We show that there is an effective uniform bound for the size of the Brauer group of $U$ in terms of the degree of $k$ .
</p>projecteuclid.org/euclid.mmj/1550480562_20190218040317Mon, 18 Feb 2019 04:03 ESTOn Unipotent Radicals of Pseudo-Reductive Groupshttps://projecteuclid.org/euclid.mmj/1550480563<strong>Michael Bate</strong>, <strong>Benjamin Martin</strong>, <strong>Gerhard Röhrle</strong>, <strong>David I. Stewart</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 23 pages.</p><p><strong>Abstract:</strong><br/>
We establish some results on the structure of the geometric unipotent radicals of pseudo-reductive $k$ -groups. In particular, our main theorem gives bounds on the nilpotency class of geometric unipotent radicals of standard pseudo-reductive groups, which are sharp in many cases. A major part of the proof rests upon consideration of the following situation: let $k'$ be a purely inseparable field extension of $k$ of degree $p^{e}$ , and let $G$ denote the Weil restriction of scalars $\mathrm{R}_{k'/k}(G')$ of a reductive $k'$ -group $G'$ . When $G=\mathrm {R}_{k'/k}(G')$ , we also provide some results on the orders of elements of the unipotent radical $\mathscr{R}_{u}(G_{\bar{k}})$ of the extension of scalars of $G$ to the algebraic closure $\bar{k}$ of $k$ .
</p>projecteuclid.org/euclid.mmj/1550480563_20190218040317Mon, 18 Feb 2019 04:03 ESTArtin Motives, Weights, and Motivic Nearby Sheaveshttps://projecteuclid.org/euclid.mmj/1551258026<strong>Florian Ivorra</strong>, <strong>Julien Sebag</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 40 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we compute the Artin part of a relative cohomological motive, introduced by Ayoub and Zucker, as a “weight zero part” in two challenging contexts. For this, we first introduce, in a very natural way, the part of punctual weight $\leqslant0$ of any complex of mixed Hodge modules and verify that the Hodge realization of the Artin part of smooth cohomological motives coincide with the part of punctual weight $\leqslant0$ of its realization. Second, we compute the Artin part of the motivic nearby sheaf, introduced by Ayoub, and relate it to the Betti cohomology of Berkovich spaces defined by tubes in non-Archimedean geometry. In particular, the former result provides a motivic interpretation of the Betti cohomology of the analytic Milnor fiber (seen as a Berkovich space).
</p>projecteuclid.org/euclid.mmj/1551258026_20190227040114Wed, 27 Feb 2019 04:01 ESTCharacterizations of Some Properties on Weighted Modulation and Wiener Amalgam Spaceshttps://projecteuclid.org/euclid.mmj/1552442712<strong>Weichao Guo</strong>, <strong>Jiecheng Chen</strong>, <strong>Dashan Fan</strong>, <strong>Guoping Zhao</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 32 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we characterize some properties on weighted modulation and Wiener amalgam spaces by the corresponding properties on weighted Lebesgue spaces. As applications, we obtain sharp conditions for product inequalities, convolution inequalities, and embedding on weighted modulation and Wiener amalgam spaces. By a unified approach different from others we give a complete answer to the question of finding sharp conditions of certain relations on weighted modulation and Wiener amalgam spaces.
</p>projecteuclid.org/euclid.mmj/1552442712_20190312220535Tue, 12 Mar 2019 22:05 EDTRandom Manifolds Have No Totally Geodesic Submanifoldshttps://projecteuclid.org/euclid.mmj/1555034652<strong>Thomas Murphy</strong>, <strong>Frederick Wilhelm</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 2, 323--335.</p><p><strong>Abstract:</strong><br/>
For $n\geq 4$ , we show that generic closed Riemannian $n$ -manifolds have no nontrivial totally geodesic submanifolds, answering a question of Spivak. Although the result is widely believed to be true, we are not aware of any proof in the literature.
</p>projecteuclid.org/euclid.mmj/1555034652_20190611220046Tue, 11 Jun 2019 22:00 EDTOn Multiplicative Dependence of Values of Rational Functions and a Generalization of the Northcott Theoremhttps://projecteuclid.org/euclid.mmj/1556589745<strong>Alina Ostafe</strong>, <strong>Min Sha</strong>, <strong>Igor E. Shparlinski</strong>, <strong>Umberto Zannier</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 2, 385--407.</p><p><strong>Abstract:</strong><br/>
In this paper, we study multiplicative dependence of values of polynomials or rational functions over a number field. As an application, we obtain new results on multiplicative dependence in the orbits of a univariate polynomial dynamical system. We also obtain a generalization of the Northcott theorem replacing the finiteness of preperiodic points from a given number field by the finiteness of algebraic integers having two multiplicatively dependent elements in their orbits.
</p>projecteuclid.org/euclid.mmj/1556589745_20190611220046Tue, 11 Jun 2019 22:00 EDTInfinitely Generated Symbolic Rees Rings of Space Monomial Curves Having Negative Curveshttps://projecteuclid.org/euclid.mmj/1557475399<strong>Kazuhiko Kurano</strong>, <strong>Koji Nishida</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 2, 409--445.</p><p><strong>Abstract:</strong><br/>
In this paper, we study finite generation of symbolic Rees rings of the defining ideal ${\mathfrak{p}}$ of the space monomial curve $(t^{a},t^{b},t^{c})$ for pairwise coprime integers $a$ , $b$ , $c$ . Suppose that the base field is of characteristic $0$ , and the ideal ${\mathfrak{p}}$ is minimally generated by three polynomials. In Theorem 1.1, under the assumption that the homogeneous element $\xi $ of the minimal degree in ${\mathfrak{p}}$ is a negative curve, we determine the minimal degree of an element $\eta $ such that the pair $\{\xi ,\eta \}$ satisfies Huneke’s criterion in the case where the symbolic Rees ring is Noetherian. By this result we can decide whether the symbolic Rees ring $\mathcal{R}_{s}({\mathfrak{p}})$ is Notherian using computers. We give a necessary and sufficient condition for finite generation of the symbolic Rees ring of ${\mathfrak{p}}$ in Proposition 4.10 under some assumptions. We give an example of an infinitely generated symbolic Rees ring of ${\mathfrak{p}}$ in which the homogeneous element of the minimal degree in ${\mathfrak{p}}^{(2)}$ is a negative curve in Example 5.7. We give a simple proof to (generalized) Huneke’s criterion.
</p>projecteuclid.org/euclid.mmj/1557475399_20190611220046Tue, 11 Jun 2019 22:00 EDTOn Separable Higher Gauss Mapshttps://projecteuclid.org/euclid.mmj/1555574416<strong>Katsuhisa Furukawa</strong>, <strong>Atsushi Ito</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 3, 483--503.</p><p><strong>Abstract:</strong><br/>
We study the $m$ th Gauss map in the sense of F. L. Zak of a projective variety $X\subset\mathbb{P}^{N}$ over an algebraically closed field in any characteristic. For all integers $m$ with $n:=\dim(X)\leqslant m\lt N$ , we show that the contact locus on $X$ of a general tangent $m$ -plane is a linear variety if the $m$ th Gauss map is separable. We also show that for smooth $X$ with $n\lt N-2$ , the $(n+1)$ th Gauss map is birational if it is separable, unless $X$ is the Segre embedding $\mathbb{P}^{1}\times\mathbb{P}^{n}\subset\mathbb{P}^{2n-1}$ . This is related to Ein’s classification of varieties with small dual varieties in characteristic zero.
</p>projecteuclid.org/euclid.mmj/1555574416_20190807040041Wed, 07 Aug 2019 04:00 EDTNoncommutative Holomorphic Semicocycleshttps://projecteuclid.org/euclid.mmj/1557302432<strong>Mark Elin</strong>, <strong>Fiana Jacobzon</strong>, <strong>Guy Katriel</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 3, 505--526.</p><p><strong>Abstract:</strong><br/>
In this paper, we study holomorphic semicocycles over semigroups in the unit disk, which take values in an arbitrary unital Banach algebra. We prove that every such semicocycle is the solution to a corresponding evolution problem. We then investigate the linearization problem: which semicocycles are cohomologous to constant semicocycles? In contrast with the case of commutative semicocycles, in the noncommutative case nonlinearizable semicocycles are shown to exist. We derive simple conditions for linearizability and show that they are sharp.
</p>projecteuclid.org/euclid.mmj/1557302432_20190807040041Wed, 07 Aug 2019 04:00 EDTMixed Weak Estimates of Sawyer Type for Commutators of Generalized Singular Integrals and Related Operatorshttps://projecteuclid.org/euclid.mmj/1559894545<strong>Fabio Berra</strong>, <strong>Marilina Carena</strong>, <strong>Gladis Pradolini</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 3, 527--564.</p><p><strong>Abstract:</strong><br/>
We study mixed weak-type inequalities for the commutator $[b,T]$ , where $b$ is a BMO function, and $T$ is a Calderón–Zygmund operator. More precisely, we prove that, for every $t\gt 0$ ,
\[uv(\{x\in\mathbb{R}^{n}:\vert \frac{[b,T](fv)(x)}{v(x)}\vert \gt t\})\leq C\int_{\mathbb{R}^{n}}\Phi (\frac{|f(x)|}{t})u(x)v(x)\,dx,\] where $\Phi(t)=t(1+\log^{+}{t})$ , $u\in A_{1}$ , and $v\in A_{\infty}(u)$ . Our technique involves the classical Calderón–Zygmund decomposition, which allows us to give a direct proof without taking into account the associated maximal operator. We use this result to prove an analogous inequality for higher-order commutators.
For a given Young function $\phi $ we also consider singular integral operators $T$ whose kernels satisfy a $L^{\phi }$ -Hörmander property, and we find sufficient conditions on $\phi $ such that a mixed weak estimate holds for $T$ and also for its higher order commutators $T^{m}_{b}$ .
We also obtain a mixed estimation for a wide class of maximal operators associated to certain Young functions of $L\log L$ type which are in intimate relation with the commutators. This last estimate involves an arbitrary weight $u$ and a radial function $v$ which is not even locally integrable.
</p>projecteuclid.org/euclid.mmj/1559894545_20190807040041Wed, 07 Aug 2019 04:00 EDTOn Certain Complex Projective Manifolds with Hodge Numbers $h^{10}=4$ and $h^{20}=5$https://projecteuclid.org/euclid.mmj/1562032917<strong>Chad Schoen</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 3, 565--596.</p>projecteuclid.org/euclid.mmj/1562032917_20190807040041Wed, 07 Aug 2019 04:00 EDTEstimation of Deviation for Random Covariance Matriceshttps://projecteuclid.org/euclid.mmj/1559894544<strong>Tien-Cuong Dinh</strong>, <strong>Duc-Viet Vu</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 3, 597--620.</p><p><strong>Abstract:</strong><br/>
We give a deviation estimate for the empirical spectral distribution of random covariance matrices whose entries are independent random variables with mean 0, variance 1, and controlled fourth moments. We also give some new properties of Laguerre polynomials.
</p>projecteuclid.org/euclid.mmj/1559894544_20190807040041Wed, 07 Aug 2019 04:00 EDTFurther Evaluation of Wahl Vanishing Theorems for Surface Singularities in Characteristic $p$https://projecteuclid.org/euclid.mmj/1560391418<strong>Masayuki Hirokado</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 3, 621--636.</p><p><strong>Abstract:</strong><br/>
Let $(\operatorname{Spec}R,\mathfrak{m})$ be a rational double point defined over an algebraically closed field $k$ of characteristic $p\geq 0$ . We evaluate further the dimensions of the local cohomology groups, which were treated by Wahl in 1975 as vanishing theorem C (resp., D) under the assumption that $p$ is a very good prime (resp., good prime) with respect to $(\operatorname{Spec}R,\mathfrak{m})$ . We use Artin’s classification of rational double points and completely determine the dimensions $\dim _{k}H_{E}^{1}(S_{X})$ and $\dim _{k}H_{E}^{1}(S_{X}\otimes\mathcal{O}_{X}(E))$ , supplementing Wahl’s theorems. In the proof, we concretely construct derivations that do not lift to the minimal resolution $X\to \operatorname{Spec}R$ and an equisingular family that injects into a versal deformation of the rational double point $(\operatorname{Spec}R,\mathfrak{m})$ .
</p>projecteuclid.org/euclid.mmj/1560391418_20190807040041Wed, 07 Aug 2019 04:00 EDTComplementary Legs and Rational Ballshttps://projecteuclid.org/euclid.mmj/1561708817<strong>Ana G. Lecuona</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 3, 637--649.</p><p><strong>Abstract:</strong><br/>
In this note, we study the Seifert rational homology spheres with two complementary legs, that is, with a pair of invariants whose fractions add up to one. We give a complete classification of the Seifert manifolds with three exceptional fibers and two complementary legs that bound rational homology balls. The result translates into a statement on the sliceness of some Montesinos knots.
</p>projecteuclid.org/euclid.mmj/1561708817_20190807040041Wed, 07 Aug 2019 04:00 EDTThe Picard Group of the Universal Abelian Variety and the Franchetta Conjecture for Abelian Varietieshttps://projecteuclid.org/euclid.mmj/1564106669<strong>Roberto Fringuelli</strong>, <strong>Roberto Pirisi</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 3, 651--671.</p><p><strong>Abstract:</strong><br/>
We compute the Picard group of the universal Abelian variety over the moduli stack $\mathscr{A}_{g,n}$ of principally polarized Abelian varieties over $\mathbb{C}$ with a symplectic principal level $n$ -structure. We then prove that over $\mathbb{C}$ the statement of the Franchetta conjecture holds in a suitable form for $\mathscr{A}_{g,n}$ .
</p>projecteuclid.org/euclid.mmj/1564106669_20190807040041Wed, 07 Aug 2019 04:00 EDT