Expand all Collapse all | Results 51 - 75 of 153 |
51. CJM 2010 (vol 62 pp. 1131)
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.xdvi Categories:14C05, qqqqq14D22, 14F05, 14J10, 14H50, 14B10, 13D02, 13D07 |
52. CJM 2010 (vol 62 pp. 1246)
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 class Categories:14M15, 14N35 |
53. 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, flag Categories:44A35, 14M15, 26D20 |
54. CJM 2010 (vol 62 pp. 668)
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 |
55. CJM 2010 (vol 62 pp. 787)
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 number Categories:14H05, 11R58, 14H45, 11G20, 11G30, 11R16, 11R29 |
56. CJM 2009 (vol 62 pp. 262)
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 |
57. CJM 2009 (vol 62 pp. 473)
GoreskyâMacPherson 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 |
58. CJM 2009 (vol 62 pp. 456)
The ChowlaâSelberg Formula and The Colmez Conjecture In this paper, we reinterpret the Colmez conjecture on the Faltings height of CM abelian varieties in terms of Hilbert (and Siegel) modular forms. We construct an elliptic modular form involving the Faltings height of a CM abelian surface and arithmetic intersection numbers, and prove that the Colmez conjecture for CM abelian surfaces is equivalent to the cuspidality of this modular form.
Categories:11G15, 11F41, 14K22 |
59. CJM 2009 (vol 61 pp. 1407)
Traces, Cross-Ratios and 2-Generator Subgroups of $\SU(2,1)$ In this work, we investigate how to decompose a pair $(A,B)$ of
loxodromic isometries of the complex hyperbolic plane $\mathbf H^{2}_{\mathbb C}$ under
the form $A=I_1I_2$ and $B=I_3I_2$, where the $I_k$'s are
involutions. The main result is a decomposability criterion, which
is expressed in terms of traces of elements of the group $\langle
A,B\rangle$.
Categories:14L24, 22E40, 32M15, 51M10 |
60. CJM 2009 (vol 61 pp. 1118)
Petits points d'une surface Pour toute sous-vari\'et\'e g\'eom\'etriquement irr\'eductible $V$
du grou\-pe multiplicatif
$\mathbb{G}_m^n$, on sait qu'en dehors d'un nombre fini de
translat\'es de tores exceptionnels
inclus dans $V$, tous les points sont de hauteur minor\'ee par une
certaine quantit\'e $q(V)^{-1}>0$. On conna\^it de plus une borne
sup\'erieure pour la somme des degr\'es de ces translat\'es de
tores pour des valeurs de $q(V)$ polynomiales en le degr\'e de $V$.
Ceci n'est pas le cas si l'on exige une minoration quasi-optimale
pour la hauteur des points de $V$, essentiellement lin\'eaire en l'inverse du degr\'e.
Nous apportons ici une r\'eponse partielle \`a ce probl\`eme\,: nous
donnons une majoration de la somme des degr\'es de ces translat\'es de
sous-tores de codimension $1$ d'une hypersurface $V$. Les r\'esultats,
obtenus dans le cas de $\mathbb{G}_m^3$, mais compl\`etement
explicites, peuvent toutefois s'\'etendre \`a $\mathbb{G}_m^n$,
moyennant quelques petites complications inh\'erentes \`a la dimension
$n$.
Keywords:Hauteur normalisÃ©e, groupe multiplicatif, problÃ¨me de Lehmer, petits points Categories:11G50, 11J81, 14G40 |
61. CJM 2009 (vol 61 pp. 1050)
Examples of Calabi--Yau 3-Folds of $\mathbb{P}^{7}$ with $\rho=1$ We give some examples of Calabi--Yau $3$-folds with $\rho=1$ and
$\rho=2$, defined over $\mathbb{Q}$ and constructed as
$4$-codimensional subvarieties of $\mathbb{P}^7$ via commutative
algebra methods. We explain how to deduce their Hodge diamond and
top Chern classes from computer based computations over some
finite field $\mathbb{F}_{p}$. Three of our examples (of degree
$17$ and $20$) are new. The two others (degree $15$ and $18$) are
known, and we recover their well-known invariants with our
method. These examples are built out of Gulliksen--Neg{\aa}rd and
Kustin--Miller complexes of locally free sheaves.
Finally, we give two new examples of Calabi--Yau $3$-folds of
$\mathbb{P}^6$ of degree $14$ and $15$ (defined over
$\mathbb{Q}$). We show that they are not deformation equivalent to
Tonoli's examples of the same degree, despite the fact that they
have the same invariants $(H^3,c_2\cdot H, c_3)$ and $\rho=1$.
Categories:14J32, 14Q15 |
62. CJM 2009 (vol 61 pp. 828)
Twisted Gross--Zagier Theorems The theorems of Gross--Zagier and Zhang relate the N\'eron--Tate
heights of complex multiplication points on the modular curve $X_0(N)$
(and on Shimura curve analogues) with the central derivatives of
automorphic $L$-function. We extend these results to include certain
CM points on modular curves of the form
$X(\Gamma_0(M)\cap\Gamma_1(S))$ (and on Shimura curve analogues).
These results are motivated by applications to Hida theory
that can be found in the companion article
"Central derivatives of $L$-functions in Hida families", Math.\ Ann.\
\textbf{399}(2007), 803--818.
Categories:11G18, 14G35 |
63. CJM 2009 (vol 61 pp. 930)
Prolongations and Computational Algebra We explore the geometric notion of prolongations in the setting of
computational algebra, extending results of Landsberg and Manivel
which relate prolongations to equations for secant varieties. We also
develop methods for computing prolongations that are combinatorial in
nature. As an application, we use prolongations to derive a new
family of secant equations for the binary symmetric model in
phylogenetics.
Categories:13P10, 14M99 |
64. CJM 2009 (vol 61 pp. 351)
Multiplication of Polynomials on Hermitian Symmetric spaces and Littlewood--Richardson Coefficients Let $K$ be a complex reductive algebraic group and $V$ a
representation of $K$. Let $S$ denote the ring of polynomials on
$V$. Assume that the action of $K$ on $S$ is multiplicity-free. If
$\lambda$ denotes the isomorphism class of an irreducible
representation of $K$, let $\rho_\lambda\from K \rightarrow
GL(V_{\lambda})$ denote the corresponding irreducible representation
and $S_\lambda$ the $\lambda$-isotypic component of $S$. Write
$S_\lambda \cdot S_\mu$ for the subspace of $S$ spanned by products of
$S_\lambda$ and $S_\mu$. If $V_\nu$ occurs as an irreducible
constituent of $V_\lambda\otimes V_\mu$, is it true that
$S_\nu\subseteq S_\lambda\cdot S_\mu$? In this paper, the authors
investigate this question for representations arising in the context
of Hermitian symmetric pairs. It is shown that the answer is yes in
some cases and, using an earlier result of Ruitenburg, that in the
remaining classical cases, the answer is yes provided that a
conjecture of Stanley on the multiplication of Jack polynomials is
true. It is also shown how the conjecture connects multiplication in
the ring $S$ to the usual Littlewood--Richardson rule.
Keywords:Hermitian symmetric spaces, multiplicity free actions, Littlewood--Richardson coefficients, Jack polynomials Categories:14L30, 22E46 |
65. CJM 2009 (vol 61 pp. 29)
The Minimal Resolution Conjecture for Points on the Cubic Surface In this paper we prove that a generalized version of the Minimal
Resolution Conjecture given by Musta\c{t}\v{a} holds for certain
general sets of points on a smooth cubic surface $X \subset
\PP^3$. The main tool used is Gorenstein liaison theory and, more
precisely, the relationship between the free resolutions of two linked schemes.
Categories:13D02, 13C40, 14M05, 14M07 |
66. CJM 2009 (vol 61 pp. 205)
Representations of Non-Negative Polynomials, Degree Bounds and Applications to Optimization Natural sufficient conditions for a polynomial to have a local minimum
at a point are considered. These conditions tend to hold with
probability $1$. It is shown that polynomials satisfying these
conditions at each minimum point have nice presentations in terms of
sums of squares. Applications are given to optimization on a compact
set and also to global optimization. In many cases, there are degree
bounds for such presentations. These bounds are of theoretical
interest, but they appear to be too large to be of much practical use
at present. In the final section, other more concrete degree bounds
are obtained which ensure at least that the feasible set of solutions
is not empty.
Categories:13J30, 12Y05, 13P99, 14P10, 90C22 |
67. CJM 2009 (vol 61 pp. 109)
The Ample Cone of the Kontsevich Moduli Space We produce ample (resp.\ NEF, eventually free) divisors in the
Kontsevich space $\Kgnb{0,n} (\mathbb P^r, d)$ of $n$-pointed,
genus $0$, stable maps to $\mathbb P^r$, given such divisors in
$\Kgnb{0,n+d}$. We prove that this produces all ample (resp.\ NEF,
eventually free) divisors in $\Kgnb{0,n}(\mathbb P^r,d)$.
As a consequence, we construct a contraction of the boundary
$\bigcup_{k=1}^{\lfloor d/2 \rfloor} \Delta_{k,d-k}$ in
$\Kgnb{0,0}(\mathbb P^r,d)$, analogous to a contraction of
the boundary $\bigcup_{k=3}^{\lfloor n/2 \rfloor}
\tilde{\Delta}_{k,n-k}$ in $\kgnb{0,n}$ first constructed by Keel
and McKernan.
Categories:14D20, 14E99, 14H10 |
68. CJM 2009 (vol 61 pp. 3)
Connected Components of Moduli Stacks of Torsors via Tamagawa Numbers Let $X$ be a smooth projective geometrically connected curve over
a finite field with function field $K$. Let $\G$ be a connected semisimple group
scheme over $X$. Under certain hypotheses we prove the equality of
two numbers associated with $\G$.
The first is an arithmetic invariant, its Tamagawa number. The second
is a geometric invariant, the number of connected components of the moduli
stack of $\G$-torsors on $X$. Our results are most useful for studying
connected components as much is known about Tamagawa numbers.
Categories:11E, 11R, 14D, 14H |
69. CJM 2008 (vol 60 pp. 1267)
Nonadjacent Radix-$\tau$ Expansions of Integers in Euclidean Imaginary Quadratic Number Fields In his seminal papers, Koblitz proposed curves
for cryptographic use. For fast operations on these curves,
these papers also
initiated a study of the radix-$\tau$ expansion of integers in the number
fields $\Q(\sqrt{-3})$ and $\Q(\sqrt{-7})$. The (window)
nonadjacent form of $\tau$-expansion of integers in
$\Q(\sqrt{-7})$ was first investigated by Solinas.
For integers in $\Q(\sqrt{-3})$, the nonadjacent form
and the window nonadjacent form of the $\tau$-expansion were
studied. These are used for efficient
point multiplications on Koblitz curves.
In this paper, we complete
the picture by producing the (window)
nonadjacent radix-$\tau$ expansions
for integers in all Euclidean imaginary quadratic number fields.
Keywords:algebraic integer, radix expression, window nonadjacent expansion, algorithm, point multiplication of elliptic curves, cryptography Categories:11A63, 11R04, 11Y16, 11Y40, 14G50 |
70. CJM 2008 (vol 60 pp. 961)
About the Defectivity of Certain Segre--Veronese Varieties We study the regularity of the higher secant varieties of $\PP^1\times
\PP^n$, embedded with divisors of type $(d,2)$ and $(d,3)$. We
produce, for the highest defective cases, a ``determinantal'' equation
of the secant variety. As a corollary, we prove that the Veronese
triple embedding of $\PP^n$ is not Grassmann defective.
Keywords:Waring problem, Segre--Veronese embedding, secant variety, Grassmann defectivity Categories:14N15, 14N05, 14M12 |
71. CJM 2008 (vol 60 pp. 875)
A Characterization of the Quantum Cohomology Ring of $G/B$ and Applications We observe that the small quantum product of the
generalized flag manifold $G/B$ is a product operation $\star$ on
$H^*(G/B)\otimes \bR[q_1,\dots, q_l]$ uniquely determined by the
facts
that: it is a deformation of the cup product on $H^*(G/B)$; it is
commutative, associative, and graded with respect to $\deg(q_i)=4$; it
satisfies a certain relation (of degree two); and the corresponding
Dubrovin connection is flat. Previously, we proved that these
properties alone imply the presentation of the ring $(H^*(G/B)\otimes
\bR[q_1,\dots, q_l],\star)$ in terms of generators and relations. In
this paper we use the above observations to give conceptually new
proofs of other fundamental results of the quantum Schubert calculus
for $G/B$: the quantum Chevalley formula of D. Peterson (see also
Fulton and Woodward ) and the ``quantization by standard
monomials" formula of Fomin, Gelfand, and Postnikov for
$G=\SL(n,\bC)$. The main idea of the proofs is the same as in
Amarzaya--Guest: from the quantum $\D$-module of $G/B$ one can
decode all information about the quantum cohomology of this space.
Categories:14M15, 14N35 |
72. CJM 2008 (vol 60 pp. 734)
Genus 2 Curves with Quaternionic Multiplication We explicitly construct the canonical rational models of Shimura
curves, both analytically in terms of modular forms and
algebraically in terms of coefficients of genus 2 curves, in the
cases of quaternion algebras of discriminant 6 and 10. This emulates
the classical construction in the elliptic curve case. We also give
families of genus 2 QM curves, whose Jacobians are the corresponding
abelian surfaces on the Shimura curve, and with coefficients that
are modular forms of weight 12. We apply these results to show
that our $j$-functions are supported exactly at those primes where
the genus 2 curve does not admit potentially good reduction, and
construct fields where this potentially good reduction is attained.
Finally, using $j$, we construct the fields of moduli and definition
for some moduli problems associated to the Atkin--Lehner group
actions.
Keywords:Shimura curve, canonical model, quaternionic multiplication, modular form, field of moduli Categories:11G18, 14G35 |
73. CJM 2008 (vol 60 pp. 532)
Local Bounds for Torsion Points on Abelian Varieties We say that an abelian variety over a $p$-adic field $K$ has
anisotropic reduction (AR) if the special fiber of its N\'eron minimal
model does not contain a nontrivial split torus. This includes all
abelian varieties with potentially good reduction and, in particular,
those with complex or quaternionic multiplication. We give a bound for
the size of the $K$-rational torsion subgroup of a $g$-dimensional AR
variety depending only on $g$ and the numerical invariants of $K$ (the
absolute ramification index and the cardinality of the residue
field). Applying these bounds to abelian varieties over a number field
with everywhere locally anisotropic reduction, we get bounds which, as
a function of $g$, are close to optimal. In particular, we determine
the possible cardinalities of the torsion subgroup of an AR abelian
surface over the rational numbers, up to a set of 11 values which are
not known to occur. The largest such value is 72.
Categories:11G10, 14K15 |
74. CJM 2008 (vol 60 pp. 556)
Polarization of Separating Invariants We prove a characteristic free version of Weyl's theorem on
polarization. Our result is an exact analogue of Weyl's theorem, the
difference being that our statement is about separating invariants
rather than generating invariants. For the special case of finite
group actions we introduce the concept of \emph{cheap polarization},
and show that it is enough to take cheap polarizations of invariants
of just one copy of a representation to obtain separating vector
invariants for any number of copies. This leads to upper bounds on
the number and degrees of separating vector invariants of finite
groups.
Keywords:Jan Draisma, Gregor Kemper, David Wehlau Categories:13A50, 14L24 |
75. CJM 2008 (vol 60 pp. 297)
Transitive Factorizations in the Hyperoctahedral Group The classical Hurwitz enumeration problem has a presentation in terms of
transitive factorizations in the symmetric group. This presentation suggests
a generalization from type~$A$ to other
finite reflection groups and, in particular, to type~$B$.
We study this generalization both from a combinatorial and a geometric
point of view, with the prospect of providing a means of understanding more
of the structure of the moduli spaces of maps with an $\gS_2$-symmetry.
The type~$A$ case has been well studied and connects Hurwitz numbers
to the moduli space of curves. We conjecture an analogous setting for the
type~$B$ case that is studied here.
Categories:05A15, 14H10, 58D29 |