Expand all Collapse all | Results 1 - 25 of 115 |
1. CMB Online first
Refined Motivic Dimension We define a refined motivic dimension for an algebraic variety
by modifying the definition of motivic dimension by Arapura.
We apply this to check and recheck the generalized Hodge conjecture
for certain varieties, such as uniruled, rationally connected
varieties and a rational surface fibration.
Keywords:motivic dimension, generalized Hodge conjecture Categories:14C30, 14C25 |
2. CMB Online first
$L$-functions for Quadratic Characters and Annihilation of Motivic Cohomology Groups Let $n$ be a positive even integer, and let $F$ be a totally real
number field and $L$ be an abelian Galois extension which is totally
real or CM.
Fix a finite set $S$ of primes of $F$ containing the infinite primes
and all those which ramify in
$L$, and let $S_L$ denote the primes of $L$ lying above those in
$S$. Then $\mathcal{O}_L^S$ denotes the ring of $S_L$-integers of $L$.
Suppose that $\psi$ is a quadratic character of the Galois group of
$L$ over $F$. Under the assumption of the motivic Lichtenbaum
conjecture, we obtain a non-trivial annihilator of the motivic
cohomology group
$H_\mathcal{M}^2(\mathcal{O}_L^S,\mathbb{Z}(n))$ from the lead term of the Taylor series for the
$S$-modified Artin $L$-function $L_{L/F}^S(s,\psi)$ at $s=1-n$.
Keywords:motivic cohomology, regulator, Artin L-functions Categories:11R42, 11R70, 14F42, 19F27 |
3. CMB 2014 (vol 58 pp. 80)
The Equivariant Cohomology Rings of Peterson Varieties in All Lie
Types Let $G$ be a complex semisimple linear algebraic group and let
$Pet$ be the associated Peterson variety in the flag
variety $G/B$.
The main theorem of this note gives an efficient presentation
of the equivariant cohomology ring $H^*_S(Pet)$ of the
Peterson variety as a quotient of a polynomial ring by an ideal
$J$ generated by quadratic polynomials, in the spirit of the
Borel presentation of the cohomology of the flag variety. Here
the group $S \cong \mathbb{C}^*$ is a certain subgroup of a maximal
torus $T$ of $G$.
Our description of the ideal $J$ uses the Cartan matrix and is
uniform across Lie types. In our arguments we use the Monk formula
and Giambelli formula for the equivariant cohomology rings of
Peterson varieties for all Lie types, as obtained in the work
of Drellich. Our result generalizes a previous theorem of Fukukawa-Harada-Masuda,
which was only for Lie type $A$.
Keywords:equivariant cohomology, Peterson varieties, flag varieties, Monk formula, Giambelli formula Categories:55N91, 14N15 |
4. CMB Online first
Homological Planes in the Grothendieck Ring of Varieties In this note, we identify, in the Grothendieck group of complex
varieties $K_0(\mathrm Var_\mathbf{C})$, the classes of $\mathbf{Q}$-homological
planes. Precisely, we prove that a connected smooth affine complex
algebraic surface $X$ is a $\mathbf{Q}$-homological plane if
and only if $[X]=[\mathbf{A}^2_\mathbf{C}]$ in the ring $K_0(\mathrm Var_\mathbf{C})$
and $\mathrm{Pic}(X)_\mathbf{Q}:=\mathrm{Pic}(X)\otimes_\mathbf{Z}\mathbf{Q}=0$.
Keywords:motivic nearby cycles, motivic Milnor fiber, nearby motives Categories:14E05, 14R10 |
5. CMB Online first
Lifting Divisors on a Generic Chain of Loops Let $C$ be a curve over a complete valued field with infinite
residue field whose skeleton is a chain of loops with generic
edge lengths. We prove that
any divisor on the chain of loops that is rational over the value
group lifts to a divisor of the same rank on $C$, confirming
a conjecture of Cools,
Draisma, Robeva, and the third author.
Keywords:tropical geometry, Brill-Noether theory, special divisors on algebraic curves Categories:14T05, 14H51 |
6. CMB 2014 (vol 57 pp. 749)
Geometric Perspective on Piecewise Polynomiality of Double Hurwitz Numbers We describe double Hurwitz numbers as intersection numbers on the
moduli space of curves $\overline{\mathcal{M}}_{g,n}$. Using a result on the
polynomiality of intersection numbers of psi classes with the Double
Ramification Cycle, our formula explains the polynomiality in chambers
of double Hurwitz numbers, and the wall crossing phenomenon in terms
of a variation of correction terms to the $\psi$ classes. We
interpret this as suggestive evidence for polynomiality of the Double
Ramification Cycle (which is only known in genera $0$ and $1$).
Keywords:double Hurwitz numbers, wall crossings, moduli spaces, ELSV formula Category:14N35 |
7. CMB 2014 (vol 57 pp. 658)
Admissibility of Local Systems for some Classes of Line Arrangements Let $\mathcal{A}$ be a line arrangement in the complex
projective plane $\mathbb{P}^2$ and let $M$ be its complement. A rank one
local system $\mathcal{L}$ on $M$ is admissible if roughly speaking
the cohomology groups
$H^m(M,\mathcal{L})$ can be computed directly from the cohomology
algebra $H^{*}(M,\mathbb{C})$. In this work, we give a sufficient
condition for the admissibility of all rank one local systems on
$M$. As a result, we obtain some properties of the characteristic
variety $\mathcal{V}_1(M)$ and the Resonance variety $\mathcal{R}_1(M)$.
Keywords:admissible local system, line arrangement, characteristic variety, multinet, resonance variety Categories:14F99, 32S22, 52C35, 05A18, 05C40, 14H50 |
8. CMB 2013 (vol 57 pp. 562)
Note on the Grothendieck Group of Subspaces of Rational Functions and Shokurov's Cartier b-divisors In a previous paper the authors developed an intersection theory for
subspaces of rational functions on an algebraic variety $X$
over $\mathbf{k} = \mathbb{C}$. In this short note, we first extend this intersection
theory to an arbitrary algebraically closed ground field $\mathbf{k}$.
Secondly we give an isomorphism between the group of Cartier
$b$-divisors on the birational class of $X$
and the Grothendieck group
of the semigroup of subspaces of rational functions on $X$. The
constructed isomorphism moreover
preserves the intersection numbers. This provides an alternative point
of view on Cartier $b$-divisors and their intersection theory.
Keywords:intersection number, Cartier divisor, Cartier b-divisor, Grothendieck group Categories:14C20, 14Exx |
9. CMB 2013 (vol 57 pp. 614)
A Note on the Weierstrass Preparation Theorem in Quasianalytic Local Rings Consider quasianalytic local rings of germs of smooth functions closed
under composition, implicit equation, and monomial division. We show
that if the Weierstrass Preparation Theorem holds in such a ring then
all elements of it are germs of analytic functions.
Categories:26E10, 26E05, 14P15 |
10. CMB 2013 (vol 57 pp. 439)
The Fixed Point Locus of the Verschiebung on $\mathcal{M}_X(2, 0)$ for Genus-2 Curves $X$ in Charateristic $2$ |
The Fixed Point Locus of the Verschiebung on $\mathcal{M}_X(2, 0)$ for Genus-2 Curves $X$ in Charateristic $2$ We prove that for every ordinary genus-$2$ curve $X$ over a finite
field $\kappa$ of characteristic $2$ with
$\textrm{Aut}(X/\kappa)=\mathbb{Z}/2\mathbb{Z} \times S_3$, there exist
$\textrm{SL}(2,\kappa[\![s]\!])$-representations of $\pi_1(X)$ such
that the image of $\pi_1(\overline{X})$ is infinite. This result
produces a family of examples similar to Laszlo's counterexample
to de Jong's question regarding the finiteness of the geometric
monodromy of representations of the fundamental group.
Keywords:vector bundle, Frobenius pullback, representation, etale fundamental group Categories:14H60, 14D05, 14G15 |
11. CMB 2012 (vol 57 pp. 97)
Rationality and the Jordan-Gatti-Viniberghi decomposition We verify
our earlier conjecture
and use it to prove that the
semisimple parts of the rational Jordan-Kac-Vinberg decompositions of
a rational vector all lie in a single rational orbit.
Keywords:reductive group, $G$-module, Jordan decomposition, orbit closure, rationality Categories:20G15, 14L24 |
12. CMB 2012 (vol 57 pp. 303)
Octonion Algebras over Rings are not Determined by their Norms Answering a question of H. Petersson, we provide
a class of examples of pair of octonion algebras over a ring having isometric
norms.
Keywords:octonion algebras, torsors, descent Categories:14L24, 20G41 |
13. CMB 2012 (vol 56 pp. 640)
Regulator Indecomposable Cycles on a Product of Elliptic Curves We provide a novel proof of the existence
of regulator indecomposables in the cycle group $CH^2(X,1)$,
where $X$ is a sufficiently general product of two elliptic
curves. In particular, the nature of our proof provides an illustration of
Beilinson rigidity.
Keywords:real regulator, regulator indecomposable, higher Chow group, indecomposable cycle Category:14C25 |
14. CMB 2011 (vol 56 pp. 225)
On the Notion of Visibility of Torsors Let $J$ be an abelian variety and
$A$ be an abelian subvariety of $J$, both defined over $\mathbf{Q}$.
Let $x$ be an element of $H^1(\mathbf{Q},A)$.
Then there are at least two definitions of $x$ being visible in $J$:
one asks that the torsor corresponding to $x$ be isomorphic over $\mathbf{Q}$
to a subvariety of $J$, and the other asks that $x$ be in the kernel
of the natural map $H^1(\mathbf{Q},A) \to H^1(\mathbf{Q},J)$. In this article, we
clarify the relation between the two definitions.
Keywords:torsors, principal homogeneous spaces, visibility, Shafarevich-Tate group Categories:11G35, 14G25 |
15. CMB 2011 (vol 56 pp. 500)
The Lang--Weil Estimate for Cubic Hypersurfaces An improved estimate is provided for the number of $\mathbb{F}_q$-rational points
on a geometrically irreducible, projective, cubic hypersurface that is
not equal to a cone.
Keywords:cubic hypersurface, rational points, finite fields Categories:11G25, 14G15 |
16. CMB 2011 (vol 55 pp. 842)
The Rank of Jacobian Varieties over the Maximal Abelian Extensions of Number Fields: Towards the Frey-Jarden Conjecture |
The Rank of Jacobian Varieties over the Maximal Abelian Extensions of Number Fields: Towards the Frey-Jarden Conjecture Frey and Jarden asked if
any abelian variety over a number field $K$
has the infinite Mordell-Weil rank over
the maximal abelian extension $K^{\operatorname{ab}}$.
In this paper,
we give an affirmative answer to their conjecture
for the Jacobian variety
of any smooth projective curve $C$
over $K$
such that $\sharp C(K^{\operatorname{ab}})=\infty$
and for any abelian variety of $\operatorname{GL}_2$-type with trivial character.
Keywords:Mordell-Weil rank, Jacobian varieties, Frey-Jarden conjecture, abelian points Categories:11G05, 11D25, 14G25, 14K07 |
17. CMB 2011 (vol 55 pp. 752)
Approximation of Holomorphic Solutions of a System of Real Analytic Equations We prove the existence of an approximation function for holomorphic
solutions of a system of real analytic equations. For this we use
ultraproducts and Weierstrass systems introduced by J. Denef and L.
Lipshitz. We also prove a version of the PÅoski smoothing theorem in
this case.
Keywords:Artin approximation, real analytic equations Categories:13B40, 13L05, 14F12 |
18. CMB 2011 (vol 55 pp. 850)
Character Sums with Division Polynomials We obtain nontrivial estimates of quadratic character sums of division polynomials $\Psi_n(P)$, $n=1,2, \dots$, evaluated at a given point $P$ on an elliptic curve over a finite field of $q$ elements. Our bounds are nontrivial if the order of $P$ is at least $q^{1/2 + \varepsilon}$ for some fixed $\varepsilon > 0$. This work is motivated by an open question about statistical indistinguishability of some cryptographically relevant sequences that was recently brought up by K. Lauter and the second author.
Keywords:division polynomial, character sum Categories:11L40, 14H52 |
19. CMB 2011 (vol 55 pp. 799)
Manifolds Covered by Lines and Extremal Rays Let $X$ be a smooth complex projective variety, and let $H \in
\operatorname{Pic}(X)$
be an ample line bundle. Assume that $X$ is covered by rational
curves with degree one with respect to $H$ and with anticanonical
degree greater than or equal to $(\dim X -1)/2$. We prove that there
is a covering family of such curves whose numerical class spans an
extremal ray in the cone of curves $\operatorname{NE}(X)$.
Keywords:rational curves, extremal rays Categories:14J40, 14E30, 14C99 |
20. CMB 2011 (vol 55 pp. 319)
The Verdier Hypercovering Theorem This note gives a simple cocycle-theoretic proof of the Verdier
hypercovering theorem. This theorem approximates morphisms $[X,Y]$ in the
homotopy category of simplicial sheaves or presheaves by simplicial
homotopy classes of maps, in the case where $Y$ is locally fibrant. The
statement proved in this paper is a generalization of the standard
Verdier hypercovering result in that it is pointed (in a very broad
sense) and there is no requirement for the source object $X$ to be
locally fibrant.
Keywords:simplicial presheaf, hypercover, cocycle Categories:14F35, 18G30, 55U35 |
21. CMB 2011 (vol 55 pp. 26)
A Mahler Measure of a $K3$ Surface Expressed as a Dirichlet $L$-Series We present another example of a $3$-variable polynomial defining a $K3$-hypersurface
and having a logarithmic Mahler measure expressed in terms of a Dirichlet
$L$-series.
Keywords:modular Mahler measure, Eisenstein-Kronecker series, $L$-series of $K3$-surfaces, $l$-adic representations, LivnÃ© criterion, Rankin-Cohen brackets Categories:11, 14D, 14J |
22. CMB 2011 (vol 54 pp. 430)
Complete Families of Linearly Non-degenerate Rational Curves We prove that every complete family of linearly non-degenerate
rational curves of degree $e > 2$ in $\mathbb{P}^{n}$ has at most $n-1$
moduli. For $e = 2$ we prove that such a family has at most $n$
moduli. The general method involves exhibiting a map from the base of
a family $X$ to the Grassmannian of $e$-planes in $\mathbb{P}^{n}$ and
analyzing the resulting map on cohomology.
Categories:14N05, 14H10 |
23. CMB 2011 (vol 54 pp. 472)
A Semiregularity Map Annihilating Obstructions to Deforming Holomorphic Maps We study infinitesimal deformations of holomorphic maps of
compact, complex, KÃ¤hler manifolds. In particular, we describe a
generalization of Bloch's semiregularity map that annihilates
obstructions to deform holomorphic maps with fixed codomain.
Keywords:semiregularity map, obstruction theory, functors of Artin rings, differential graded Lie algebras Categories:13D10, 14D15, 14B10 |
24. CMB 2010 (vol 54 pp. 561)
A Note on Toric Varieties Associated with Moduli Spaces
In this note we give a brief review of the construction of a toric
variety $\mathcal{V}$ coming from a genus $g \geq 2$ Riemann surface
$\Sigma^g$ equipped with a trinion, or pair of pants, decomposition.
This was outlined by J. Hurtubise and L.~C. Jeffrey.
A. Tyurin used this construction on a certain
collection of trinion decomposed surfaces to produce a variety
$DM_g$, the so-called \emph{Delzant model of moduli space}, for
each genus $g.$ We conclude this note with some basic facts about
the moment polytopes of the varieties $\mathcal{V}.$ In particular,
we show that the varieties $DM_g$ constructed by Tyurin, and claimed
to be smooth, are in fact singular for $g \geq 3.$
Categories:14M25, 52B20 |
25. CMB 2010 (vol 54 pp. 520)
Simple Helices on Fano Threefolds
Building on the work of Nogin,
we prove that the braid group $B_4$ acts transitively on full exceptional
collections of vector bundles on Fano threefolds with $b_2=1$ and
$b_3=0$. Equivalently,
this group acts transitively on the set of simple helices (considered
up to a shift in the derived category) on such a Fano threefold. We
also prove that on
threefolds with $b_2=1$ and very ample anticanonical class, every
exceptional coherent
sheaf is locally free.
Categories:14F05, 14J45 |