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51. CJM 2011 (vol 64 pp. 3)

Boissière, Samuel
 Automorphismes naturels de l'espace de Douady de points sur une surface On Ã©tablit quelques rÃ©sultats gÃ©nÃ©raux relatifs Ã  la taille du groupe d'automorphismes de l'espace de Douady de points sur une surface, puis on Ã©tudie quelques propriÃ©tÃ©s des automorphismes provenant d'un automorphisme de la surface, en particulier leur action sur la cohomologie et la classification de leurs points fixes. Keywords:SchÃ©ma de Hilbert, automorphismes, points fixesCategory:14C05

52. CJM 2011 (vol 64 pp. 1122)

 $p$-adic $L$-functions and the Rationality of Darmon Cycles Darmon cycles are a higher weight analogue of Stark--Heegner points. They yield local cohomology classes in the Deligne representation associated with a cuspidal form on $\Gamma _{0}( N)$ of even weight $k_{0}\geq 2$. They are conjectured to be the restriction of global cohomology classes in the Bloch--Kato Selmer group defined over narrow ring class fields attached to a real quadratic field. We show that suitable linear combinations of them obtained by genus characters satisfy these conjectures. We also prove $p$-adic Gross--Zagier type formulas, relating the derivatives of $p$-adic $L$-functions of the weight variable attached to imaginary (resp. real) quadratic fields to Heegner cycles (resp. Darmon cycles). Finally we express the second derivative of the Mazur--Kitagawa $p$-adic $L$-function of the weight variable in terms of a global cycle defined over a quadratic extension of $\mathbb{Q}$. Categories:11F67, 14G05

53. CJM 2011 (vol 64 pp. 845)

Helm, David; Katz, Eric
 Monodromy Filtrations and the Topology of Tropical Varieties We study the topology of tropical varieties that arise from a certain natural class of varieties. We use the theory of tropical degenerations to construct a natural, multiplicity-free'' parameterization of $\operatorname{Trop}(X)$ by a topological space $\Gamma_X$ and give a geometric interpretation of the cohomology of $\Gamma_X$ in terms of the action of a monodromy operator on the cohomology of $X$. This gives bounds on the Betti numbers of $\Gamma_X$ in terms of the Betti numbers of $X$ which constrain the topology of $\operatorname{Trop}(X)$. We also obtain a description of the top power of the monodromy operator acting on middle cohomology of $X$ in terms of the volume pairing on $\Gamma_X$. Categories:14T05, 14D06

54. CJM 2011 (vol 64 pp. 805)

Chapon, François; Defosseux, Manon
 Quantum Random Walks and Minors of Hermitian Brownian Motion Considering quantum random walks, we construct discrete-time approximations of the eigenvalues processes of minors of Hermitian Brownian motion. It has been recently proved by Adler, Nordenstam, and van Moerbeke that the process of eigenvalues of two consecutive minors of a Hermitian Brownian motion is a Markov process; whereas, if one considers more than two consecutive minors, the Markov property fails. We show that there are analog results in the noncommutative counterpart and establish the Markov property of eigenvalues of some particular submatrices of Hermitian Brownian motion. Keywords:quantum random walk, quantum Markov chain, generalized casimir operators, Hermitian Brownian motion, diffusions, random matrices, minor processCategories:46L53, 60B20, 14L24

55. CJM 2011 (vol 64 pp. 123)

Lee, Jae-Hyouk
 Gosset Polytopes in Picard Groups of del Pezzo Surfaces In this article, we study the correspondence between the geometry of del Pezzo surfaces $S_{r}$ and the geometry of the $r$-dimensional Gosset polytopes $(r-4)_{21}$. We construct Gosset polytopes $(r-4)_{21}$ in $\operatorname{Pic} S_{r}\otimes\mathbb{Q}$ whose vertices are lines, and we identify divisor classes in $\operatorname{Pic} S_{r}$ corresponding to $(a-1)$-simplexes ($a\leq r$), $(r-1)$-simplexes and $(r-1)$-crosspolytopes of the polytope $(r-4)_{21}$. Then we explain how these classes correspond to skew $a$-lines($a\leq r$), exceptional systems, and rulings, respectively. As an application, we work on the monoidal transform for lines to study the local geometry of the polytope $(r-4)_{21}$. And we show that the Gieser transformation and the Bertini transformation induce a symmetry of polytopes $3_{21}$ and $4_{21}$, respectively. Categories:51M20, 14J26, 22E99

56. CJM 2011 (vol 63 pp. 1345)

Jardine, J. F.
 Pointed Torsors This paper gives a characterization of homotopy fibres of inverse image maps on groupoids of torsors that are induced by geometric morphisms, in terms of both pointed torsors and pointed cocycles, suitably defined. Cocycle techniques are used to give a complete description of such fibres, when the underlying geometric morphism is the canonical stalk on the classifying topos of a profinite group $G$. If the torsors in question are defined with respect to a constant group $H$, then the path components of the fibre can be identified with the set of continuous maps from the profinite group $G$ to the group $H$. More generally, when $H$ is not constant, this set of path components is the set of continuous maps from a pro-object in sheaves of groupoids to $H$, which pro-object can be viewed as a Grothendieck fundamental groupoid". Keywords:pointed torsors, pointed cocycles, homotopy fibresCategories:18G50, 14F35, 55B30

57. CJM 2011 (vol 64 pp. 409)

Rainer, Armin
 Lifting Quasianalytic Mappings over Invariants Let $\rho \colon G \to \operatorname{GL}(V)$ be a rational finite dimensional complex representation of a reductive linear algebraic group $G$, and let $\sigma_1,\dots,\sigma_n$ be a system of generators of the algebra of invariant polynomials $\mathbb C[V]^G$. We study the problem of lifting mappings $f\colon \mathbb R^q \supseteq U \to \sigma(V) \subseteq \mathbb C^n$ over the mapping of invariants $\sigma=(\sigma_1,\dots,\sigma_n) \colon V \to \sigma(V)$. Note that $\sigma(V)$ can be identified with the categorical quotient $V /\!\!/ G$ and its points correspond bijectively to the closed orbits in $V$. We prove that if $f$ belongs to a quasianalytic subclass $\mathcal C \subseteq C^\infty$ satisfying some mild closedness properties that guarantee resolution of singularities in $\mathcal C$, e.g., the real analytic class, then $f$ admits a lift of the same class $\mathcal C$ after desingularization by local blow-ups and local power substitutions. As a consequence we show that $f$ itself allows for a lift that belongs to $\operatorname{SBV}_{\operatorname{loc}}$, i.e., special functions of bounded variation. If $\rho$ is a real representation of a compact Lie group, we obtain stronger versions. Keywords:lifting over invariants, reductive group representation, quasianalytic mappings, desingularization, bounded variationCategories:14L24, 14L30, 20G20, 22E45

58. CJM 2011 (vol 64 pp. 81)

David, C.; Wu, J.
 Pseudoprime Reductions of Elliptic Curves Let $E$ be an elliptic curve over $\mathbb Q$ without complex multiplication, and for each prime $p$ of good reduction, let $n_E(p) = | E(\mathbb F_p) |$. For any integer $b$, we consider elliptic pseudoprimes to the base $b$. More precisely, let $Q_{E,b}(x)$ be the number of primes $p \leq x$ such that $b^{n_E(p)} \equiv b\,({\rm mod}\,n_E(p))$, and let $\pi_{E, b}^{\operatorname{pseu}}(x)$ be the number of compositive $n_E(p)$ such that $b^{n_E(p)} \equiv b\,({\rm mod}\,n_E(p))$ (also called elliptic curve pseudoprimes). Motivated by cryptography applications, we address the problem of finding upper bounds for $Q_{E,b}(x)$ and $\pi_{E, b}^{\operatorname{pseu}}(x)$, generalising some of the literature for the classical pseudoprimes to this new setting. Keywords:Rosser-Iwaniec sieve, group order of elliptic curves over finite fields, pseudoprimes Categories:11N36, 14H52

59. CJM 2011 (vol 63 pp. 1058)

Easton, Robert W.
 $S_3$-covers of Schemes We analyze flat $S_3$-covers of schemes, attempting to create structures parallel to those found in the abelian and triple cover theories. We use an initial local analysis as a guide in finding a global description. Keywords:nonabelian groups, permutation group, group covers, schemesCategory:14L30

60. CJM 2011 (vol 63 pp. 992)

Bruin, Nils; Doerksen, Kevin
 The Arithmetic of Genus Two Curves with (4,4)-Split Jacobians In this paper we study genus $2$ curves whose Jacobians admit a polarized $(4,4)$-isogeny to a product of elliptic curves. We consider base fields of characteristic different from $2$ and $3$, which we do not assume to be algebraically closed. We obtain a full classification of all principally polarized abelian surfaces that can arise from gluing two elliptic curves along their $4$-torsion, and we derive the relation their absolute invariants satisfy. As an intermediate step, we give a general description of Richelot isogenies between Jacobians of genus $2$ curves, where previously only Richelot isogenies with kernels that are pointwise defined over the base field were considered. Our main tool is a Galois theoretic characterization of genus $2$ curves admitting multiple Richelot isogenies. Keywords:Genus 2 curves, isogenies, split Jacobians, elliptic curvesCategories:11G30, 14H40

61. CJM 2011 (vol 63 pp. 1388)

Misamore, Michael D.
 Nonabelian $H^1$ and the Ãtale Van Kampen Theorem Generalized Ã©tale homotopy pro-groups $\pi_1^{\operatorname{Ã©t}}(Ä{C}, x)$ associated with pointed, connected, small Grothendieck sites $(\mathcal{C}, x)$ are defined, and their relationship to Galois theory and the theory of pointed torsors for discrete groups is explained. Applications include new rigorous proofs of some folklore results around $\pi_1^{\operatorname{Ã©t}}(Ã©t(X), x)$, a description of Grothendieck's short exact sequence for Galois descent in terms of pointed torsor trivializations, and a new Ã©tale van Kampen theorem that gives a simple statement about a pushout square of pro-groups that works for covering families that do not necessarily consist exclusively of monomorphisms. A corresponding van Kampen result for Grothendieck's profinite groups $\pi_1^{\mathrm{Gal}}$ immediately follows. Keywords:Ã©tale homotopy theory, simplicial sheavesCategories:18G30, 14F35

62. CJM 2011 (vol 63 pp. 755)

Chu, Kenneth C. K.
 On the Geometry of the Moduli Space of Real Binary Octics The moduli space of smooth real binary octics has five connected components. They parametrize the real binary octics whose defining equations have $0,\dots,4$ complex-conjugate pairs of roots respectively. We show that each of these five components has a real hyperbolic structure in the sense that each is isomorphic as a real-analytic manifold to the quotient of an open dense subset of $5$-dimensional real hyperbolic space $\mathbb{RH}^5$ by the action of an arithmetic subgroup of $\operatorname{Isom}(\mathbb{RH}^5)$. These subgroups are commensurable to discrete hyperbolic reflection groups, and the Vinberg diagrams of the latter are computed. Keywords:real binary octics, moduli space, complex hyperbolic geometry, Vinberg algorithmCategories:32G13, 32G20, 14D05, 14D20

63. CJM 2011 (vol 63 pp. 878)

Howard, Benjamin; Manon, Christopher; Millson, John
 The Toric Geometry of Triangulated Polygons in Euclidean Spac Speyer and Sturmfels associated GrÃ¶bner toric degenerations $\mathrm{Gr}_2(\mathbb{C}^n)^{\mathcal{T}}$ of $\mathrm{Gr}_2(\mathbb{C}^n)$ with each trivalent tree $\mathcal{T}$ having $n$ leaves. These degenerations induce toric degenerations $M_{\mathbf{r}}^{\mathcal{T}}$ of $M_{\mathbf{r}}$, the space of $n$ ordered, weighted (by $\mathbf{r}$) points on the projective line. Our goal in this paper is to give a geometric (Euclidean polygon) description of the toric fibers and describe the action of the compact part of the torus as "bendings of polygons". We prove the conjecture of Foth and Hu that the toric fibers are homeomorphic to the spaces defined by Kamiyama and Yoshida. Categories:14L24, 53D20

64. CJM 2011 (vol 63 pp. 616)

Lee, Edward
 A Modular Quintic Calabi-Yau Threefold of Level 55 In this note we search the parameter space of Horrocks-Mumford quintic threefolds and locate a Calabi-Yau threefold that is modular, in the sense that the $L$-function of its middle-dimensional cohomology is associated with a classical modular form of weight 4 and level 55. Keywords: Calabi-Yau threefold, non-rigid Calabi-Yau threefold, two-dimensional Galois representation, modular variety, Horrocks-Mumford vector bundleCategories:14J15, 11F23, 14J32, 11G40

65. CJM 2011 (vol 63 pp. 481)

Baragar, Arthur
 The Ample Cone for a K3 Surface In this paper, we give several pictorial fractal representations of the ample or KÃ¤hler cone for surfaces in a certain class of $K3$ surfaces. The class includes surfaces described by smooth $(2,2,2)$ forms in ${\mathbb P^1\times\mathbb P^1\times \mathbb P^1}$ defined over a sufficiently large number field $K$ that have a line parallel to one of the axes and have Picard number four. We relate the Hausdorff dimension of this fractal to the asymptotic growth of orbits of curves under the action of the surface's group of automorphisms. We experimentally estimate the Hausdorff dimension of the fractal to be $1.296 \pm .010$. Keywords:Fractal, Hausdorff dimension, K3 surface, Kleinian groups, dynamicsCategories:14J28, , , , 14J50, 11D41, 11D72, 11H56, 11G10, 37F35, 37D05

66. CJM 2010 (vol 63 pp. 86)

Chen, Xi
 On Vojta's $1+\varepsilon$ Conjecture We give another proof of Vojta's $1+\varepsilon$ conjecture. Keywords:Vojta, 1+epsilonCategories:14G40, 14H15

67. CJM 2010 (vol 62 pp. 1293)

Kasprzyk, Alexander M.
 Canonical Toric Fano Threefolds An inductive approach to classifying all toric Fano varieties is given. As an application of this technique, we present a classification of the toric Fano threefolds with at worst canonical singularities. Up to isomorphism, there are $674,\!688$ such varieties. Keywords:toric, Fano, threefold, canonical singularities, convex polytopesCategories:14J30, 14J30, 14M25, 52B20

68. CJM 2010 (vol 62 pp. 1201)

Alzati, Alberto; Besana, Gian Mario
 Criteria for Very Ampleness of Rank Two Vector Bundles over Ruled Surfaces Very ampleness criteria for rank $2$ vector bundles over smooth, ruled surfaces over rational and elliptic curves are given. The criteria are then used to settle open existence questions for some special threefolds of low degree. Keywords:vector bundles, very ampleness, ruled surfacesCategories:14E05, 14J30

69. CJM 2010 (vol 62 pp. 1131)

Kleppe, Jan O.
 Moduli Spaces of Reflexive Sheaves of Rank 2 Let $\mathcal{F}$ be a coherent rank $2$ sheaf on a scheme $Y \subset \mathbb{P}^{n}$ of dimension at least two and let $X \subset Y$ be the zero set of a section $\sigma \in H^0(\mathcal{F})$. In this paper, we study the relationship between the functor that deforms the pair $(\mathcal{F},\sigma)$ and the two functors that deform $\mathcal{F}$ on $Y$, and $X$ in $Y$, respectively. By imposing some conditions on two forgetful maps between the functors, we prove that the scheme structure of \emph{e.g.,} the moduli scheme ${\rm M_Y}(P)$ of stable sheaves on a threefold $Y$ at $(\mathcal{F})$, and the scheme structure at $(X)$ of the Hilbert scheme of curves on $Y$ become closely related. Using this relationship, we get criteria for the dimension and smoothness of ${\rm M_{Y}}(P)$ at $(\mathcal{F})$, without assuming ${\textrm{Ext}^2}(\mathcal{F} ,\mathcal{F} ) = 0$. For reflexive sheaves on $Y=\mathbb{P}^{3}$ whose deficiency module $M = H_{*}^1(\mathcal{F})$ satisfies ${_{0}\! \textrm{Ext}^2}(M ,M ) = 0$ (\emph{e.g.,} of diameter at most 2), we get necessary and sufficient conditions of unobstructedness that coincide in the diameter one case. The conditions are further equivalent to the vanishing of certain graded Betti numbers of the free graded minimal resolution of $H_{*}^0(\mathcal{F})$. Moreover, we show that every irreducible component of ${\rm M}_{\mathbb{P}^{3}}(P)$ containing a reflexive sheaf of diameter one is reduced (generically smooth) and we compute its dimension. We also determine a good lower bound for the dimension of any component of ${\rm M}_{\mathbb{P}^{3}}(P)$ that contains a reflexive stable sheaf with small'' deficiency module $M$. Keywords:moduli space, reflexive sheaf, Hilbert scheme, space curve, Buchsbaum sheaf, unobstructedness, cup product, graded Betti numbers.xdviCategories:14C05, qqqqq14D22, 14F05, 14J10, 14H50, 14B10, 13D02, 13D07

70. CJM 2010 (vol 62 pp. 1246)

Chaput, P. E.; Manivel, L.; Perrin, N.
 Quantum Cohomology of Minuscule Homogeneous Spaces III. Semi-Simplicity and Consequences We prove that the quantum cohomology ring of any minuscule or cominuscule homogeneous space, specialized at $q=1$, is semisimple. This implies that complex conjugation defines an algebra automorphism of the quantum cohomology ring localized at the quantum parameter. We check that this involution coincides with the strange duality defined in our previous article. We deduce Vafa--Intriligator type formulas for the Gromov--Witten invariants. Keywords:quantum cohomology, minuscule homogeneous spaces, Schubert calculus, quantum Euler classCategories:14M15, 14N35

71. CJM 2010 (vol 62 pp. 870)

 The Brascamp-Lieb Polyhedron A set of necessary and sufficient conditions for the Brascamp--Lieb inequality to hold has recently been found by Bennett, Carbery, Christ, and Tao. We present an analysis of these conditions. This analysis allows us to give a concise description of the set where the inequality holds in the case where each of the linear maps involved has co-rank $1$. This complements the result of Barthe concerning the case where the linear maps all have rank $1$. Pushing our analysis further, we describe the case where the maps have either rank $1$ or rank $2$. A separate but related problem is to give a list of the finite number of conditions necessary and sufficient for the Brascamp--Lieb inequality to hold. We present an algorithm which generates such a list. Keywords:Brascamp-Lieb inequality, Loomis-Whitney inequality, lattice, flagCategories:44A35, 14M15, 26D20

72. CJM 2010 (vol 62 pp. 668)

Vollaard, Inken
 The Supersingular Locus of the Shimura Variety for GU(1,s) In this paper we study the supersingular locus of the reduction modulo $p$ of the Shimura variety for $GU(1,s)$ in the case of an inert prime $p$. Using DieudonnÃ© theory we define a stratification of the corresponding moduli space of $p$-divisible groups. We describe the incidence relation of this stratification in terms of the Bruhat--Tits building of a unitary group. In the case of $GU(1,2)$, we show that the supersingular locus is equidimensional of dimension 1 and is of complete intersection. We give an explicit description of the irreducible components and their intersection behaviour. Categories:14G35, 11G18, 14K10

73. CJM 2010 (vol 62 pp. 787)

Landquist, E.; Rozenhart, P.; Scheidler, R.; Webster, J.; Wu, Q.
 An Explicit Treatment of Cubic Function Fields with Applications We give an explicit treatment of cubic function fields of characteristic at least five. This includes an efficient technique for converting such a field into standard form, formulae for the field discriminant and the genus, simple necessary and sufficient criteria for non-singularity of the defining curve, and a characterization of all triangular integral bases. Our main result is a description of the signature of any rational place in a cubic extension that involves only the defining curve and the order of the base field. All these quantities only require simple polynomial arithmetic as well as a few square-free polynomial factorizations and, in some cases, square and cube root extraction modulo an irreducible polynomial. We also illustrate why and how signature computation plays an important role in computing the class number of the function field. This in turn has applications to the study of zeros of zeta functions of function fields. Keywords:cubic function field, discriminant, non-singularity, integral basis, genus, signature of a place, class numberCategories:14H05, 11R58, 14H45, 11G20, 11G30, 11R16, 11R29

74. CJM 2009 (vol 62 pp. 262)

Goresky, Mark; MacPherson, Robert
 On the Spectrum of the Equivariant Cohomology Ring If an algebraic torus $T$ acts on a complex projective algebraic variety $X$, then the affine scheme $\operatorname{Spec} H^*_T(X;\mathbb C)$ associated with the equivariant cohomology is often an arrangement of linear subspaces of the vector space $H_2^T(X;\mathbb C).$ In many situations the ordinary cohomology ring of $X$ can be described in terms of this arrangement. Categories:14L30, 54H15

75. CJM 2009 (vol 62 pp. 473)

Yun, Zhiwei
 GoreskyâMacPherson Calculus for the Affine Flag Varieties We use the fixed point arrangement technique developed by Goresky and MacPherson to calculate the part of the equivariant cohomology of the affine flag variety $\mathcal{F}\ell_G$ generated by degree 2. We use this result to show that the vertices of the moment map image of $\mathcal{F}\ell_G$ lie on a paraboloid. Categories:14L30, 55N91
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