Algebra & Number Theory Articles (Project Euclid)
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The latest articles from Algebra & Number Theory on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2017 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 19 Oct 2017 12:49 EDTThu, 19 Oct 2017 12:49 EDThttps://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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The umbral moonshine module for the unique unimodular Niemeier root system
https://projecteuclid.org/euclid.ant/1508431771
<strong>John Duncan</strong>, <strong>Jeffrey Harvey</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 11, Number 3, 505--535.</p><p><strong>Abstract:</strong><br/>
We use canonically twisted modules for a certain super vertex operator algebra to construct the umbral moonshine module for the unique Niemeier lattice that coincides with its root sublattice. In particular, we give explicit expressions for the vector-valued mock modular forms attached to automorphisms of this lattice by umbral moonshine. We also characterize the vector-valued mock modular forms arising, in which four of Ramanujan’s fifth-order mock theta functions appear as components.
</p>projecteuclid.org/euclid.ant/1508431771_20171019124943Thu, 19 Oct 2017 12:49 EDTInfinitely generated symbolic Rees algebras over finite fieldshttps://projecteuclid.org/euclid.ant/1572314507<strong>Akiyoshi Sannai</strong>, <strong>Hiromu Tanaka</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 8, 1879--1891.</p><p><strong>Abstract:</strong><br/>
For the polynomial ring over an arbitrary field with twelve variables, there exists a prime ideal whose symbolic Rees algebra is not finitely generated.
</p>projecteuclid.org/euclid.ant/1572314507_20191028220203Mon, 28 Oct 2019 22:02 EDTManin's $b$-constant in familieshttps://projecteuclid.org/euclid.ant/1572314508<strong>Akash Kumar Sengupta</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 8, 1893--1905.</p><p><strong>Abstract:</strong><br/>
We show that the [math] -constant (appearing in Manin’s conjecture) is constant on very general fibers of a family of algebraic varieties. If the fibers of the family are uniruled, then we show that the [math] -constant is constant on general fibers.
</p>projecteuclid.org/euclid.ant/1572314508_20191028220203Mon, 28 Oct 2019 22:02 EDTEquidimensional adic eigenvarieties for groups with discrete serieshttps://projecteuclid.org/euclid.ant/1572314509<strong>Daniel R. Gulotta</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 8, 1907--1940.</p><p><strong>Abstract:</strong><br/>
We extend Urban’s construction of eigenvarieties for reductive groups [math] such that [math] has discrete series to include characteristic [math] points at the boundary of weight space. In order to perform this construction, we define a notion of “locally analytic” functions and distributions on a locally [math] -analytic manifold taking values in a complete Tate [math] -algebra in which [math] is not necessarily invertible. Our definition agrees with the definition of locally analytic distributions on [math] -adic Lie groups given by Johansson and Newton.
</p>projecteuclid.org/euclid.ant/1572314509_20191028220203Mon, 28 Oct 2019 22:02 EDTCohomological and numerical dynamical degrees on abelian varietieshttps://projecteuclid.org/euclid.ant/1572314510<strong>Fei Hu</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 8, 1941--1958.</p><p><strong>Abstract:</strong><br/>
We show that for a self-morphism of an abelian variety defined over an algebraically closed field of arbitrary characteristic, the second cohomological dynamical degree coincides with the first numerical dynamical degree.
</p>projecteuclid.org/euclid.ant/1572314510_20191028220203Mon, 28 Oct 2019 22:02 EDTA comparison between pro-$p$ Iwahori–Hecke modules and mod $p$ representationshttps://projecteuclid.org/euclid.ant/1572314511<strong>Noriyuki Abe</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 8, 1959--1981.</p><p><strong>Abstract:</strong><br/>
We give an equivalence of categories between certain subcategories of modules of pro- [math] Iwahori–Hecke algebras and modulo [math] representations.
</p>projecteuclid.org/euclid.ant/1572314511_20191028220203Mon, 28 Oct 2019 22:02 EDTProof of a conjecture of Colliot-Thélène and a diophantine excision theoremhttps://projecteuclid.org/euclid.ant/1576292479<strong>Jan Denef</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 9, 1983--1996.</p><p><strong>Abstract:</strong><br/>
We prove a conjecture of Colliot-Thélène that implies the Ax–Kochen Theorem on [math] -adic forms. We obtain it as an easy consequence of a diophantine excision theorem whose proof forms the body of the present paper.
</p>projecteuclid.org/euclid.ant/1576292479_20191213220142Fri, 13 Dec 2019 22:01 ESTIrreducible characters with bounded root Artin conductorhttps://projecteuclid.org/euclid.ant/1576292483<strong>Amalia Pizarro-Madariaga</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 9, 1997--2004.</p><p><strong>Abstract:</strong><br/>
We prove that the best possible lower bound for the Artin conductor is exponential in the degree.
</p>projecteuclid.org/euclid.ant/1576292483_20191213220142Fri, 13 Dec 2019 22:01 ESTFrobenius–Perron theory of endofunctorshttps://projecteuclid.org/euclid.ant/1576292484<strong>Jianmin Chen</strong>, <strong>Zhibin Gao</strong>, <strong>Elizabeth Wicks</strong>, <strong>James J. Zhang</strong>, <strong>Xiaohong Zhang</strong>, <strong>Hong Zhu</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 9, 2005--2055.</p><p><strong>Abstract:</strong><br/>
We introduce the Frobenius–Perron dimension of an endofunctor of a [math] -linear category and provide some applications.
</p>projecteuclid.org/euclid.ant/1576292484_20191213220142Fri, 13 Dec 2019 22:01 ESTPositivity of anticanonical divisors and $F$-purity of fibershttps://projecteuclid.org/euclid.ant/1576292485<strong>Sho Ejiri</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 9, 2057--2080.</p><p><strong>Abstract:</strong><br/>
We prove that given a flat generically smooth morphism between smooth projective varieties with [math] - pure closed fibers, if the source space is Fano, weak Fano or a variety with nef anticanonical divisor, respectively, then so is the target space. We also show that, in arbitrary characteristic, a generically smooth surjective morphism between smooth projective varieties cannot have nef and big relative anticanonical divisor, if the target space has positive dimension.
</p>projecteuclid.org/euclid.ant/1576292485_20191213220142Fri, 13 Dec 2019 22:01 ESTA probabilistic approach to systems of parameters and Noether normalizationhttps://projecteuclid.org/euclid.ant/1576292486<strong>Juliette Bruce</strong>, <strong>Daniel Erman</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 9, 2081--2102.</p><p><strong>Abstract:</strong><br/>
We study systems of parameters over finite fields from a probabilistic perspective and use this to give the first effective Noether normalization result over a finite field. Our central technique is an adaptation of Poonen’s closed point sieve, where we sieve over higher dimensional subvarieties, and we express the desired probabilities via a zeta function-like power series that enumerates higher dimensional varieties instead of closed points. This also yields a new proof of a recent result of Gabber, Liu and Lorenzini (2015) and Chinburg, Moret-Bailly, Pappas and Taylor (2017) on Noether normalizations of projective families over the integers.
</p>projecteuclid.org/euclid.ant/1576292486_20191213220142Fri, 13 Dec 2019 22:01 ESTThe structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectureshttps://projecteuclid.org/euclid.ant/1576292487<strong>Terence Tao</strong>, <strong>Joni Teräväinen</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 9, 2103--2150.</p><p><strong>Abstract:</strong><br/>
We study the asymptotic behaviour of higher order correlations
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as a function of the parameters [math] and [math] , where [math] are bounded multiplicative functions, [math] are integer shifts, and [math] is large. Our main structural result asserts, roughly speaking, that such correlations asymptotically vanish for almost all [math] if [math] does not (weakly) pretend to be a twisted Dirichlet character [math] , and behave asymptotically like a multiple of [math] otherwise. This extends our earlier work on the structure of logarithmically averaged correlations, in which the [math] parameter is averaged out and one can set [math] . Among other things, the result enables us to establish special cases of the Chowla and Elliott conjectures for (unweighted) averages at almost all scales; for instance, we establish the [math] -point Chowla conjecture [math] for [math] odd or equal to [math] for all scales [math] outside of a set of zero logarithmic density.
</p>projecteuclid.org/euclid.ant/1576292487_20191213220142Fri, 13 Dec 2019 22:01 ESTVI-modules in nondescribing characteristic, part Ihttps://projecteuclid.org/euclid.ant/1576292488<strong>Rohit Nagpal</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 9, 2151--2189.</p><p><strong>Abstract:</strong><br/>
Let VI be the category of finite dimensional [math] -vector spaces whose morphisms are injective linear maps and let [math] be a noetherian ring. We study the category of functors from VI to [math] -modules in the case when [math] is invertible in [math] . Our results include a structure theorem, finiteness of regularity, and a description of the Hilbert series. These results are crucial in the classification of smooth irreducible [math] -representations in nondescribing characteristic which is contained in Part II of this paper ( VI-modules in nondescribing characteristic , part II, arxiv:1810.04592).
</p>projecteuclid.org/euclid.ant/1576292488_20191213220142Fri, 13 Dec 2019 22:01 ESTDegree of irrationality of very general abelian surfaceshttps://projecteuclid.org/euclid.ant/1576292489<strong>Nathan Chen</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 9, 2191--2198.</p><p><strong>Abstract:</strong><br/>
The degree of irrationality of a projective variety [math] is defined to be the smallest degree of a rational dominant map to a projective space of the same dimension. For abelian surfaces, Yoshihara computed this invariant in specific cases, while Stapleton gave a sublinear upper bound for very general polarized abelian surfaces [math] of degree [math] . Somewhat surprisingly, we show that the degree of irrationality of a very general polarized abelian surface is uniformly bounded above by 4, independently of the degree of the polarization. This result disproves part of a conjecture of Bastianelli, De Poi, Ein, Lazarsfeld, and Ullery.
</p>projecteuclid.org/euclid.ant/1576292489_20191213220142Fri, 13 Dec 2019 22:01 ESTLower bounds for the least prime in Chebotarevhttps://projecteuclid.org/euclid.ant/1576292490<strong>Andrew Fiori</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 9, 2199--2203.</p><p><strong>Abstract:</strong><br/>
In this paper we show there exists an infinite family of number fields [math] , Galois over [math] , for which the smallest prime [math] of [math] which splits completely in [math] has size at least [math] . This gives a converse to various upper bounds, which shows that they are best possible.
</p>projecteuclid.org/euclid.ant/1576292490_20191213220142Fri, 13 Dec 2019 22:01 ESTBrody hyperbolicity of base spaces of certain families of varietieshttps://projecteuclid.org/euclid.ant/1576292491<strong>Mihnea Popa</strong>, <strong>Behrouz Taji</strong>, <strong>Lei Wu</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 9, 2205--2242.</p><p><strong>Abstract:</strong><br/>
We prove that quasi-projective base spaces of smooth families of minimal varieties of general type with maximal variation do not admit Zariski dense entire curves. We deduce the fact that moduli stacks of polarized varieties of this sort are Brody hyperbolic, answering a special case of a question of Viehweg and Zuo. For two-dimensional bases, we show analogous results in the more general case of families of varieties admitting a good minimal model.
</p>projecteuclid.org/euclid.ant/1576292491_20191213220142Fri, 13 Dec 2019 22:01 ESTThe elliptic KZB connection and algebraic de Rham theory for unipotent fundamental groups of elliptic curveshttps://projecteuclid.org/euclid.ant/1579143614<strong>Ma Luo</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 10, 2243--2275.</p><p><strong>Abstract:</strong><br/>
We develop an algebraic de Rham theory for unipotent fundamental groups of once punctured elliptic curves over a field of characteristic zero using the universal elliptic KZB connection of Calaque, Enriquez and Etingof (2009) and Levin and Racinet (2007). We use it to give an explicit version of Tannaka duality for unipotent connections over an elliptic curve with a regular singular point at the identity.
</p>projecteuclid.org/euclid.ant/1579143614_20200115220030Wed, 15 Jan 2020 22:00 ESTMoments of random multiplicative functions, II: High momentshttps://projecteuclid.org/euclid.ant/1579143615<strong>Adam J Harper</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 10, 2277--2321.</p><p><strong>Abstract:</strong><br/>
We determine the order of magnitude of [math] up to factors of size [math] , where [math] is a Steinhaus or Rademacher random multiplicative function, for all real [math] .
In the Steinhaus case, we show that [math] on this whole range. In the Rademacher case, we find a transition in the behavior of the moments when [math] , where the size starts to be dominated by “orthogonal” rather than “unitary” behavior. We also deduce some consequences for the large deviations of [math] .
The proofs use various tools, including hypercontractive inequalities, to connect [math] with the [math] -th moment of an Euler product integral. When [math] is large, it is then fairly easy to analyze this integral. When [math] is close to 1 the analysis seems to require subtler arguments, including Doob’s [math] maximal inequality for martingales.
</p>projecteuclid.org/euclid.ant/1579143615_20200115220030Wed, 15 Jan 2020 22:00 ESTArtin–Mazur–Milne duality for fppf cohomologyhttps://projecteuclid.org/euclid.ant/1579143616<strong>Cyril Demarche</strong>, <strong>David Harari</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 10, 2323--2357.</p><p><strong>Abstract:</strong><br/>
We provide a complete proof of a duality theorem for the fppf cohomology of either a curve over a finite field or a ring of integers of a number field, which extends the classical Artin–Verdier Theorem in étale cohomology. We also prove some finiteness and vanishing statements.
</p>projecteuclid.org/euclid.ant/1579143616_20200115220030Wed, 15 Jan 2020 22:00 ESTBetti numbers of Shimura curves and arithmetic three-orbifoldshttps://projecteuclid.org/euclid.ant/1579143617<strong>Mikołaj Frączyk</strong>, <strong>Jean Raimbault</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 10, 2359--2382.</p><p><strong>Abstract:</strong><br/>
We show that asymptotically the first Betti number [math] of a Shimura curve satisfies the Gauss–Bonnet equality [math] where [math] is hyperbolic volume; equivalently [math] where [math] is the arithmetic genus. We also show that the first Betti number of a congruence hyperbolic 3-orbifold asymptotically vanishes relatively to hyperbolic volume, that is [math] . This generalizes previous results obtained by Frączyk, on which we rely, and uses the same main tool, namely Benjamini–Schramm convergence.
</p>projecteuclid.org/euclid.ant/1579143617_20200115220030Wed, 15 Jan 2020 22:00 ESTCombinatorial identities and Titchmarsh's divisor problem for multiplicative functionshttps://projecteuclid.org/euclid.ant/1579143618<strong>Sary Drappeau</strong>, <strong>Berke Topacogullari</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 10, 2383--2425.</p><p><strong>Abstract:</strong><br/>
Given a multiplicative function [math] which is periodic over the primes, we obtain a full asymptotic expansion for the shifted convolution sum [math] , where [math] denotes the divisor function and [math] . We consider in particular the special cases where [math] is the generalized divisor function [math] with [math] , and the characteristic function of sums of two squares (or more generally, ideal norms of abelian extensions). As another application, we deduce a full asymptotic expansion in the generalized Titchmarsh divisor problem [math] , where [math] counts the number of distinct prime divisors of [math] , thus extending a result of Fouvry and Bombieri, Friedlander and Iwaniec.
We present two different proofs: The first relies on an effective combinatorial formula of Heath-Brown’s type for the divisor function [math] with [math] , and an interpolation argument in the [math] -variable for weighted mean values of [math] . The second is based on an identity of Linnik type for [math] and the well-factorability of friable numbers.
</p>projecteuclid.org/euclid.ant/1579143618_20200115220030Wed, 15 Jan 2020 22:00 ESTThe construction problem for Hodge numbers modulo an integerhttps://projecteuclid.org/euclid.ant/1579143619<strong>Matthias Paulsen</strong>, <strong>Stefan Schreieder</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 10, 2427--2434.</p><p><strong>Abstract:</strong><br/>
For any integer [math] and any dimension [math] , we show that any [math] -dimensional Hodge diamond with values in [math] is attained by the Hodge numbers of an [math] -dimensional smooth complex projective variety. As a corollary, there are no polynomial relations among the Hodge numbers of [math] -dimensional smooth complex projective varieties besides the ones induced by the Hodge symmetries, which answers a question raised by Kollár in 2012.
</p>projecteuclid.org/euclid.ant/1579143619_20200115220030Wed, 15 Jan 2020 22:00 ESTCrystalline comparison isomorphisms in $p$-adic Hodge theory:the absolutely unramified casehttps://projecteuclid.org/euclid.ant/1579143649<strong>Fucheng Tan</strong>, <strong>Jilong Tong</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 7, 1509--1581.</p><p><strong>Abstract:</strong><br/>
We construct the crystalline comparison isomorphisms for proper smooth formal schemes over an absolutely unramified base. Such isomorphisms hold for étale cohomology with nontrivial coefficients, as well as in the relative setting, i.e., for proper smooth morphisms of smooth formal schemes. The proof is formulated in terms of the proétale topos introduced by Scholze, and uses his primitive comparison theorem for the structure sheaf on the proétale site. Moreover, we need to prove the Poincaré lemma for crystalline period sheaves, for which we adapt the idea of Andreatta and Iovita. Another ingredient for the proof is the geometric acyclicity of crystalline period sheaves, whose computation is due to Andreatta and Brinon.
</p>projecteuclid.org/euclid.ant/1579143649_20200115220053Wed, 15 Jan 2020 22:00 ESTPseudorepresentations of weight one are unramifiedhttps://projecteuclid.org/euclid.ant/1579143650<strong>Frank Calegari</strong>, <strong>Joel Specter</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 7, 1583--1596.</p><p><strong>Abstract:</strong><br/>
We prove that the determinant (pseudorepresentation) associated to the Hecke algebra of Katz modular forms of weight one and level prime to [math] is unramified at [math] .
</p>projecteuclid.org/euclid.ant/1579143650_20200115220053Wed, 15 Jan 2020 22:00 ESTOn the $p$-typical de Rham–Witt complex over $W(k)$https://projecteuclid.org/euclid.ant/1579143651<strong>Christopher Davis</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 7, 1597--1631.</p><p><strong>Abstract:</strong><br/>
Hesselholt and Madsen (2004) define and study the (absolute, [math] -typical) de Rham–Witt complex in mixed characteristic, where [math] is an odd prime. They give as an example an elementary algebraic description of the de Rham–Witt complex over [math] , [math] . The main goal of this paper is to construct, for [math] a perfect ring of characteristic [math] , a Witt complex over [math] with an algebraic description which is completely analogous to Hesselholt and Madsen’s description for [math] . Our Witt complex is not isomorphic to the de Rham–Witt complex; instead we prove that, in each level, the de Rham–Witt complex over [math] surjects onto our Witt complex, and that the kernel consists of all elements which are divisible by arbitrarily high powers of [math] . We deduce an explicit description of [math] for each [math] . We also deduce results concerning the de Rham–Witt complex over certain [math] -torsion-free perfectoid rings.
</p>projecteuclid.org/euclid.ant/1579143651_20200115220053Wed, 15 Jan 2020 22:00 ESTCoherent Tannaka duality and algebraicity of Hom-stackshttps://projecteuclid.org/euclid.ant/1579143652<strong>Jack Hall</strong>, <strong>David Rydh</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 7, 1633--1675.</p><p><strong>Abstract:</strong><br/>
We establish Tannaka duality for noetherian algebraic stacks with affine stabilizer groups. Our main application is the existence of [math] -stacks in great generality.
</p>projecteuclid.org/euclid.ant/1579143652_20200115220053Wed, 15 Jan 2020 22:00 ESTSupercuspidal representations of ${\rm GL}_n({\rm F})$ distinguished by a Galois involutionhttps://projecteuclid.org/euclid.ant/1579143653<strong>Vincent Sécherre</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 7, 1677--1733.</p><p><strong>Abstract:</strong><br/>
Let [math] be a quadratic extension of nonarchimedean locally compact fields of residual characteristic [math] and let [math] denote its nontrivial automorphism. Let [math] be an algebraically closed field of characteristic different from [math] . To any cuspidal representation [math] of [math] , with coefficients in [math] , such that [math] (such a representation is said to be [math] -selfdual) we associate a quadratic extension [math] , where [math] is a tamely ramified extension of [math] and [math] is a tamely ramified extension of [math] , together with a quadratic character of [math] . When [math] is supercuspidal, we give a necessary and sufficient condition, in terms of these data, for [math] to be [math] -distinguished. When the characteristic [math] of [math] is not [math] , denoting by [math] the nontrivial [math] -character of [math] trivial on [math] -norms, we prove that any [math] -selfdual supercuspidal [math] -representation is either distinguished or [math] -distinguished, but not both. In the modular case, that is when [math] , we give examples of [math] -selfdual cuspidal nonsupercuspidal representations which are not distinguished nor [math] -distinguished. In the particular case where [math] is the field of complex numbers, in which case all cuspidal representations are supercuspidal, this gives a complete distinction criterion for arbitrary complex cuspidal representations, as well as a purely local proof, for cuspidal representations, of the dichotomy and disjunction theorem due to Kable and Anandavardhanan, Kable and Tandon, when [math] .
</p>projecteuclid.org/euclid.ant/1579143653_20200115220053Wed, 15 Jan 2020 22:00 ESTA vanishing result for higher smooth dualshttps://projecteuclid.org/euclid.ant/1579143654<strong>Claus Sorensen</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 7, 1735--1763.</p><p><strong>Abstract:</strong><br/>
In this paper we prove a general vanishing result for Kohlhaase’s higher smooth duality functors [math] . If [math] is any unramified connected reductive [math] -adic group, [math] is a hyperspecial subgroup, and [math] is a Serre weight, we show that [math] for [math] , where [math] is a Borel subgroup and the dimension is over [math] . This is due to Kohlhaase for [math] , in which case it has applications to the calculation of [math] for supersingular representations. Our proof avoids explicit matrix computations by making use of Lazard theory, and we deduce our result from an analogous statement for graded algebras via a spectral sequence argument. The graded case essentially follows from Koszul duality between symmetric and exterior algebras.
</p>projecteuclid.org/euclid.ant/1579143654_20200115220053Wed, 15 Jan 2020 22:00 ESTGorenstein-projective and semi-Gorenstein-projective moduleshttps://projecteuclid.org/euclid.ant/1586224818<strong>Claus Michael Ringel</strong>, <strong>Pu Zhang</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 1, 1--36.</p><p><strong>Abstract:</strong><br/>
Let [math] be an artin algebra. An [math] -module [math] will be said to be semi-Gorenstein-projective provided that [math] for all [math] . All Gorenstein-projective modules are semi-Gorenstein-projective and only few and quite complicated examples of semi-Gorenstein-projective modules which are not Gorenstein-projective have been known. One of the aims of the paper is to provide conditions on [math] such that all semi-Gorenstein-projective left modules are Gorenstein-projective (we call such an algebra left weakly Gorenstein). In particular, we show that in case there are only finitely many isomorphism classes of indecomposable left modules which are both semi-Gorenstein-projective and torsionless, then [math] is left weakly Gorenstein. On the other hand, we exhibit a 6-dimensional algebra [math] with a semi-Gorenstein-projective module [math] which is not torsionless (thus not Gorenstein-projective). Actually, also the [math] -dual module [math] is semi-Gorenstein-projective. In this way, we show the independence of the total reflexivity conditions of Avramov and Martsinkovsky, thus completing a partial proof by Jorgensen and Şega. Since all the syzygy-modules of [math] and [math] are 3-dimensional, the example can be checked (and visualized) quite easily.
</p>projecteuclid.org/euclid.ant/1586224818_20200406220023Mon, 06 Apr 2020 22:00 EDTThe 16-rank of $\mathbb{Q}(\sqrt{-p})$https://projecteuclid.org/euclid.ant/1586224819<strong>Peter Koymans</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 1, 37--65.</p><p><strong>Abstract:</strong><br/>
Recently, a density result for the [math] -rank of [math] was established when [math] varies among the prime numbers, assuming a short character sum conjecture. We prove the same density result unconditionally.
</p>projecteuclid.org/euclid.ant/1586224819_20200406220023Mon, 06 Apr 2020 22:00 EDTSupersingular Hecke modules as Galois representationshttps://projecteuclid.org/euclid.ant/1586224820<strong>Elmar Grosse-Klönne</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 1, 67--118.</p><p><strong>Abstract:</strong><br/>
Let [math] be a local field of mixed characteristic [math] , let [math] be a finite extension of its residue field, let [math] be the pro- [math] -Iwahori Hecke [math] -algebra attached to [math] for some [math] . We construct an exact and fully faithful functor from the category of supersingular [math] -modules to the category of [math] -representations over [math] . More generally, for a certain [math] -algebra [math] surjecting onto [math] we define the notion of [math] -supersingular modules and construct an exact and fully faithful functor from the category of [math] -supersingular [math] -modules to the category of [math] -representations over [math] .
</p>projecteuclid.org/euclid.ant/1586224820_20200406220023Mon, 06 Apr 2020 22:00 EDTStability in the homology of unipotent groupshttps://projecteuclid.org/euclid.ant/1586224821<strong>Andrew Putman</strong>, <strong>Steven V Sam</strong>, <strong>Andrew Snowden</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 1, 119--154.</p><p><strong>Abstract:</strong><br/>
Let [math] be a (not necessarily commutative) ring whose additive group is finitely generated and let [math] be the group of upper-triangular unipotent matrices over [math] . We study how the homology groups of [math] vary with [math] from the point of view of representation stability. Our main theorem asserts that if for each [math] we have representations [math] of [math] over a ring [math] that are appropriately compatible and satisfy suitable finiteness hypotheses, then the rule [math] defines a finitely generated [math] -module. As a consequence, if [math] is a field then [math] is eventually equal to a polynomial in [math] . We also prove similar results for the Iwahori subgroups of [math] for number rings [math] .
</p>projecteuclid.org/euclid.ant/1586224821_20200406220023Mon, 06 Apr 2020 22:00 EDTOn the orbits of multiplicative pairshttps://projecteuclid.org/euclid.ant/1586224822<strong>Oleksiy Klurman</strong>, <strong>Alexander P. Mangerel</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 1, 155--189.</p><p><strong>Abstract:</strong><br/>
We characterize all pairs of completely multiplicative functions [math] , where [math] denotes the unit circle, such that
{
(
f
(
n
)
,
g
(
n
+
1
)
)
}
n
≥
1
¯
≠
𝕋
×
𝕋
.
In so doing, we settle an old conjecture of Zoltán Daróczy and Imre Kátai.
</p>projecteuclid.org/euclid.ant/1586224822_20200406220023Mon, 06 Apr 2020 22:00 EDTBirationally superrigid Fano 3-folds of codimension 4https://projecteuclid.org/euclid.ant/1586224823<strong>Takuzo Okada</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 1, 191--212.</p><p><strong>Abstract:</strong><br/>
We determine birational superrigidity for a quasismooth prime Fano [math] -fold of codimension [math] with no projection centers. In particular we prove birational superrigidity for Fano [math] -folds of codimension [math] with no projection centers which were recently constructed by Coughlan and Ducat. We also pose some questions and a conjecture regarding the classification of birationally superrigid Fano [math] -folds.
</p>projecteuclid.org/euclid.ant/1586224823_20200406220023Mon, 06 Apr 2020 22:00 EDTCoble fourfold, $\mathfrak S_6$-invariant quartic threefolds, and Wiman–Edge sexticshttps://projecteuclid.org/euclid.ant/1586224824<strong>Ivan Cheltsov</strong>, <strong>Alexander Kuznetsov</strong>, <strong>Konstantin Shramov</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 1, 213--274.</p><p><strong>Abstract:</strong><br/>
We construct two small resolutions of singularities of the Coble fourfold (the double cover of the four-dimensional projective space branched over the Igusa quartic). We use them to show that all [math] -invariant three-dimensional quartics are birational to conic bundles over the quintic del Pezzo surface with the discriminant curves from the Wiman–Edge pencil. As an application, we check that [math] -invariant three-dimensional quartics are unirational, obtain new proofs of rationality of four special quartics among them and irrationality of the others, and describe their Weil divisor class groups as [math] -representations.
</p>projecteuclid.org/euclid.ant/1586224824_20200406220023Mon, 06 Apr 2020 22:00 EDTOn the definition of quantum Heisenberg categoryhttps://projecteuclid.org/euclid.ant/1591668013<strong>Jonathan Brundan</strong>, <strong>Alistair Savage</strong>, <strong>Ben Webster</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 2, 275--321.</p><p><strong>Abstract:</strong><br/>
We introduce a diagrammatic monoidal category [math] which we call the quantum Heisenberg category ; here, [math] is “central charge” and [math] and [math] are invertible parameters. Special cases were known before: for central charge [math] and parameters [math] and [math] our quantum Heisenberg category may be obtained from the deformed version of Khovanov’s Heisenberg category introduced by Licata and Savage by inverting its polynomial generator, while [math] is the affinization of the HOMFLY-PT skein category. We also prove a basis theorem for the morphism spaces in [math] .
</p>projecteuclid.org/euclid.ant/1591668013_20200608220020Mon, 08 Jun 2020 22:00 EDTCharacteristic cycles and Gevrey series solutions of $A$-hypergeometric systemshttps://projecteuclid.org/euclid.ant/1591668014<strong>Christine Berkesch</strong>, <strong>María-Cruz Fernández-Fernández</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 2, 323--347.</p><p><strong>Abstract:</strong><br/>
We compute the [math] -characteristic cycle of an [math] -hypergeometric system and higher Euler–Koszul homology modules of the toric ring. We also prove upper semicontinuity results about the multiplicities in these cycles and apply our results to analyze the behavior of Gevrey solution spaces of the system.
</p>projecteuclid.org/euclid.ant/1591668014_20200608220020Mon, 08 Jun 2020 22:00 EDTSingularity categories of deformations of Kleinian singularitieshttps://projecteuclid.org/euclid.ant/1591668015<strong>Simon Crawford</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 2, 349--382.</p><p><strong>Abstract:</strong><br/>
Let [math] be a finite subgroup of [math] and let [math] be the coordinate ring of the corresponding Kleinian singularity. In 1998, Crawley-Boevey and Holland defined deformations [math] of [math] parametrised by weights [math] . In this paper, we determine the singularity categories [math] of these deformations, and show that they correspond to subgraphs of the Dynkin graph associated to [math] . This generalises known results on the structure of [math] . We also provide a generalisation of the intersection theory appearing in the geometric McKay correspondence to a noncommutative setting.
</p>projecteuclid.org/euclid.ant/1591668015_20200608220020Mon, 08 Jun 2020 22:00 EDTIwasawa main conjecture for Rankin–Selberg $p$-adic $L$-functionshttps://projecteuclid.org/euclid.ant/1591668016<strong>Xin Wan</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 2, 383--483.</p><p><strong>Abstract:</strong><br/>
In this paper we prove that the [math] -adic [math] -function that interpolates the Rankin–Selberg product of a general modular form and a CM form of higher weight divides the characteristic ideal of the corresponding Selmer group. This is one divisibility of the Iwasawa main conjecture for this [math] -adic [math] -function. We prove this conjecture using congruences between Klingen–Eisenstein series and cusp forms on the group [math] , following the strategy of recent work by C. Skinner and E. Urban. The actual argument is, however, more complicated due to the need to work with general Fourier–Jacobi expansions. This theorem is used to deduce a converse of the Gross–Zagier–Kolyvagin theorem and the [math] -adic part of the precise BSD formula in the rank one case.
</p>projecteuclid.org/euclid.ant/1591668016_20200608220020Mon, 08 Jun 2020 22:00 EDTPositivity results for spaces of rational curveshttps://projecteuclid.org/euclid.ant/1591668017<strong>Roya Beheshti</strong>, <strong>Eric Riedl</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 2, 485--500.</p><p><strong>Abstract:</strong><br/>
Let [math] be a very general hypersurface of degree [math] in [math] . We investigate positivity properties of the spaces [math] of degree [math] rational curves in [math] . We show that for small [math] , [math] has no rational curves meeting the locus of smooth embedded curves. We show that for [math] , there are no rational curves other than lines in the locus [math] swept out by lines. We exhibit differential forms on a smooth compactification of [math] for every [math] and [math] .
</p>projecteuclid.org/euclid.ant/1591668017_20200608220020Mon, 08 Jun 2020 22:00 EDTGeneralized Schur algebrashttps://projecteuclid.org/euclid.ant/1591668018<strong>Alexander Kleshchev</strong>, <strong>Robert Muth</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 2, 501--544.</p><p><strong>Abstract:</strong><br/>
We define and study a new class of bialgebras, which generalize certain Turner double algebras related to generic blocks of symmetric groups. Bases and generators of these algebras are given. We investigate when the algebras are symmetric which is relevant to block theory of finite groups. We then establish a double centralizer property related to blocks of Schur algebras.
</p>projecteuclid.org/euclid.ant/1591668018_20200608220020Mon, 08 Jun 2020 22:00 EDTThe distribution of $p$-torsion in degree $p$ cyclic fieldshttps://projecteuclid.org/euclid.ant/1593482419<strong>Jack Klys</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 4, 815--854.</p><p><strong>Abstract:</strong><br/>
We compute all the moments of the [math] -torsion in the first step of a filtration of the class group defined by Gerth (1987) for cyclic fields of degree [math] , unconditionally for [math] and under GRH in general. We show that it satisfies a distribution which Gerth conjectured as an extension of the Cohen–Lenstra–Martinet conjectures. In the [math] case this gives the distribution of the [math] -torsion of the class group modulo the Galois invariant part. We follow the strategy used by Fouvry and Klüners (2007) in their proof of the distribution of the [math] -torsion in quadratic fields.
</p>projecteuclid.org/euclid.ant/1593482419_20200629220025Mon, 29 Jun 2020 22:00 EDTOn the motivic class of an algebraic grouphttps://projecteuclid.org/euclid.ant/1593482420<strong>Federico Scavia</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 4, 855--866.</p><p><strong>Abstract:</strong><br/>
Let [math] be a field of characteristic zero admitting a biquadratic field extension. We give an example of a torus [math] over [math] whose classifying stack [math] is stably rational and such that [math] in the Grothendieck ring of algebraic stacks over [math] . We also give an example of a finite étale group scheme [math] over [math] such that [math] is stably rational and [math] .
</p>projecteuclid.org/euclid.ant/1593482420_20200629220025Mon, 29 Jun 2020 22:00 EDTA representation theory approach to integral moments of $L$-functions over function fieldshttps://projecteuclid.org/euclid.ant/1593482421<strong>Will Sawin</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 4, 867--906.</p><p><strong>Abstract:</strong><br/>
We propose a new heuristic approach to integral moments of [math] -functions over function fields, which we demonstrate in the case of Dirichlet characters ramified at one place (the function field analogue of the moments of the Riemann zeta function, where we think of the character [math] as ramified at the infinite place). We represent the moment as a sum of traces of Frobenius on cohomology groups associated to irreducible representations. Conditional on a hypothesis on the vanishing of some of these cohomology groups, we calculate the moments of the [math] -function and they match the predictions of the Conry, Farmer, Keating, Rubinstein, and Snaith recipe ( Proc. Lond. Math. Soc. [math] 91 (2005), 33–104).
In this case, the decomposition into irreducible representations seems to separate the main term and error term, which are mixed together in the long sums obtained from the approximate functional equation, even when it is dyadically decomposed. This makes our heuristic statement relatively simple, once the geometric background is set up. We hope that this will clarify the situation in more difficult cases like the [math] -functions of quadratic Dirichlet characters to squarefree modulus. There is also some hope for a geometric proof of this cohomological hypothesis, which would resolve the moment problem for these [math] -functions in the large degree limit over function fields.
</p>projecteuclid.org/euclid.ant/1593482421_20200629220025Mon, 29 Jun 2020 22:00 EDTDeformations of smooth complete toric varieties: obstructions and the cup producthttps://projecteuclid.org/euclid.ant/1593482422<strong>Nathan Ilten</strong>, <strong>Charles Turo</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 4, 907--926.</p><p><strong>Abstract:</strong><br/>
Let [math] be a complete [math] -factorial toric variety. We explicitly describe the space [math] and the cup product map [math] in combinatorial terms. Using this, we give an example of a smooth projective toric threefold for which the cup product map does not vanish, showing that in general, smooth complete toric varieties may have obstructed deformations.
</p>projecteuclid.org/euclid.ant/1593482422_20200629220025Mon, 29 Jun 2020 22:00 EDTMass equidistribution on the torus in the depth aspecthttps://projecteuclid.org/euclid.ant/1593482423<strong>Yueke Hu</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 4, 927--946.</p><p><strong>Abstract:</strong><br/>
In this paper we prove the equidistribution of the restriction of the mass of automorphic newforms to a nonsplit torus in the depth aspect. This result is better than the current known results on the similar problem in the eigenvalue aspect. The method is relatively elementary and makes use of the known effective QUE result in the depth aspect.
</p>projecteuclid.org/euclid.ant/1593482423_20200629220025Mon, 29 Jun 2020 22:00 EDTThe basepoint-freeness threshold and syzygies of abelian varietieshttps://projecteuclid.org/euclid.ant/1593482424<strong>Federico Caucci</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 4, 947--960.</p><p><strong>Abstract:</strong><br/>
We show how a natural constant introduced by Jiang and Pareschi for a polarized abelian variety encodes information about the syzygies of the section ring of the polarization. As a particular case this gives a quick and characteristic-free proof of Lazarsfeld’s conjecture on syzygies of abelian varieties, originally proved by Pareschi in characteristic zero.
</p>projecteuclid.org/euclid.ant/1593482424_20200629220025Mon, 29 Jun 2020 22:00 EDTOn the Ekedahl–Oort stratification of Shimura curveshttps://projecteuclid.org/euclid.ant/1593482425<strong>Benjamin Howard</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 4, 961--990.</p><p><strong>Abstract:</strong><br/>
We study the Hodge–Tate period domain associated to a quaternionic Shimura curve at a prime of bad reduction, and give an explicit description of its Ekedahl–Oort stratification.
</p>projecteuclid.org/euclid.ant/1593482425_20200629220025Mon, 29 Jun 2020 22:00 EDTA moving lemma for relative 0-cycleshttps://projecteuclid.org/euclid.ant/1593482426<strong>Amalendu Krishna</strong>, <strong>Jinhyun Park</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 4, 991--1054.</p><p><strong>Abstract:</strong><br/>
We prove a moving lemma for the additive and ordinary higher Chow groups of relative [math] -cycles of regular semilocal [math] -schemes essentially of finite type over an infinite perfect field. From this, we show that the cycle classes can be represented by cycles that possess certain finiteness, surjectivity, and smoothness properties. It plays a key role in showing that the crystalline cohomology of smooth varieties can be expressed in terms of algebraic cycles.
</p>projecteuclid.org/euclid.ant/1593482426_20200629220025Mon, 29 Jun 2020 22:00 EDTThe algebraic de Rham realization of the elliptic polylogarithm via the Poincaré bundlehttps://projecteuclid.org/euclid.ant/1593655265<strong>Johannes Sprang</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 3, 545--585.</p><p><strong>Abstract:</strong><br/>
We describe the algebraic de Rham realization of the elliptic polylogarithm for arbitrary families of elliptic curves in terms of the Poincaré bundle. Our work builds on previous work of Scheider and generalizes results of Bannai, Kobayashi and Tsuji, and Scheider. As an application, we compute the de Rham–Eisenstein classes explicitly in terms of certain algebraic Eisenstein series.
</p>projecteuclid.org/euclid.ant/1593655265_20200701220115Wed, 01 Jul 2020 22:01 EDT$a$-numbers of curves in Artin–Schreier covershttps://projecteuclid.org/euclid.ant/1593655266<strong>Jeremy Booher</strong>, <strong>Bryden Cais</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 3, 587--641.</p><p><strong>Abstract:</strong><br/>
Let [math] be a branched [math] -cover of smooth, projective, geometrically connected curves over a perfect field of characteristic [math] . We investigate the relationship between the [math] -numbers of [math] and [math] and the ramification of the map [math] . This is analogous to the relationship between the genus (respectively [math] -rank) of [math] and [math] given the Riemann–Hurwitz (respectively Deuring–Shafarevich) formula. Except in special situations, the [math] -number of [math] is not determined by the [math] -number of [math] and the ramification of the cover, so we instead give bounds on the [math] -number of [math] . We provide examples showing our bounds are sharp. The bounds come from a detailed analysis of the kernel of the Cartier operator.
</p>projecteuclid.org/euclid.ant/1593655266_20200701220115Wed, 01 Jul 2020 22:01 EDTOn the locus of $2$-dimensional crystalline representations with a given reduction modulo $p$https://projecteuclid.org/euclid.ant/1593655267<strong>Sandra Rozensztajn</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 3, 643--700.</p><p><strong>Abstract:</strong><br/>
We consider the family of irreducible crystalline representations of dimension [math] of [math] given by the [math] for a fixed weight [math] . We study the locus of the parameter [math] where these representations have a given reduction modulo [math] . We give qualitative results on this locus and show that for a fixed [math] and [math] it can be computed by determining the reduction modulo [math] of [math] for a finite number of values of the parameter [math] . We also generalize these results to other Galois types.
</p>projecteuclid.org/euclid.ant/1593655267_20200701220115Wed, 01 Jul 2020 22:01 EDTThird Galois cohomology group of function fields of curves over number fieldshttps://projecteuclid.org/euclid.ant/1593655269<strong>Venapally Suresh</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 3, 701--729.</p><p><strong>Abstract:</strong><br/>
Let [math] be a number field or a [math] -adic field and [math] the function field of a curve over [math] . Let [math] be a prime. Suppose that [math] contains a primitive [math] -th root of unity. If [math] and [math] is a number field, then assume that [math] is totally imaginary. In this article we show that every element in [math] is a symbol. This leads to the finite generation of the Chow group of zero-cycles on a quadric fibration of a curve over a totally imaginary number field.
</p>projecteuclid.org/euclid.ant/1593655269_20200701220115Wed, 01 Jul 2020 22:01 EDTOn upper bounds of Manin typehttps://projecteuclid.org/euclid.ant/1593655270<strong>Sho Tanimoto</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 3, 731--761.</p><p><strong>Abstract:</strong><br/>
We introduce a certain birational invariant of a polarized algebraic variety and use that to obtain upper bounds for the counting functions of rational points on algebraic varieties. Using our theorem, we obtain new upper bounds of Manin type for [math] deformation types of smooth Fano [math] -folds of Picard rank [math] following the Mori–Mukai classification. We also find new upper bounds for polarized K3 surfaces [math] of Picard rank [math] using Bayer and Macrì’s result on the nef cone of the Hilbert scheme of two points on [math] .
</p>projecteuclid.org/euclid.ant/1593655270_20200701220115Wed, 01 Jul 2020 22:01 EDTTubular approaches to Baker's method for curves and varietieshttps://projecteuclid.org/euclid.ant/1593655271<strong>Samuel Le Fourn</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 3, 763--785.</p><p><strong>Abstract:</strong><br/>
Baker’s method, relying on estimates on linear forms in logarithms of algebraic numbers, allows one to prove in several situations the effective finiteness of integral points on varieties. In this article, we generalize results of Levin regarding Baker’s method for varieties, and explain how, quite surprisingly, it mixes (under additional hypotheses) with Runge’s method to improve some known estimates in the case of curves by bypassing (or more generally reducing) the need for linear forms in [math] -adic logarithms. We then use these ideas to improve known estimates on solutions of [math] -unit equations. Finally, we explain how a finer analysis and formalism can improve upon the conditions given, and give some applications to the Siegel modular variety [math] .
</p>projecteuclid.org/euclid.ant/1593655271_20200701220115Wed, 01 Jul 2020 22:01 EDTFano 4-folds with rational fibrationshttps://projecteuclid.org/euclid.ant/1593655272<strong>Cinzia Casagrande</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 3, 787--813.</p><p><strong>Abstract:</strong><br/>
We study (smooth, complex) Fano 4-folds [math] having a rational contraction of fiber type, that is, a rational map [math] that factors as a sequence of flips followed by a contraction of fiber type. The existence of such a map is equivalent to the existence of a nonzero, nonbig movable divisor on [math] . Our main result is that if [math] is not [math] or [math] , then the Picard number [math] of [math] is at most 18, with equality only if [math] is a product of surfaces. We also show that if a Fano 4-fold [math] has a dominant rational map [math] , regular and proper on an open subset of [math] , with [math] , then either [math] is a product of surfaces, or [math] is at most 12. These results are part of a program to study Fano 4-folds with large Picard number via birational geometry.
</p>projecteuclid.org/euclid.ant/1593655272_20200701220115Wed, 01 Jul 2020 22:01 EDT