Expand all Collapse all | Results 26 - 50 of 115 |
26. CMB 2010 (vol 54 pp. 381)
A Short Note on the Higher Level Version of the Krull--Baer Theorem
Klep and Velu\v{s}\v{c}ek generalized the Krull--Baer theorem for
higher level preorderings to the non-commutative setting. A $n$-real valuation
$v$ on a skew field $D$ induces a group homomorphism $\overline{v}$. A section
of $\overline{v}$ is a crucial ingredient of the construction of a complete
preordering on the base field $D$ such that its projection on the residue skew
field $k_v$ equals the given level $1$ ordering on $k_v$. In the article we give
a proof of the existence of the section of $\overline{v}$, which was left as an
open problem by Klep and Velu\v{s}\v{c}ek, and thus
complete the generalization of the Krull--Baer theorem for preorderings.
Keywords:orderings of higher level, division rings, valuations Categories:14P99, 06Fxx |
27. CMB 2010 (vol 54 pp. 56)
Characteristic Varieties for a Class of Line Arrangements
Let $\mathcal{A}$ be a line arrangement in the complex projective plane
$\mathbb{P}^2$, having the points of multiplicity $\geq 3$ situated on two
lines in $\mathcal{A}$, say $H_0$ and $H_{\infty}$. Then we show that the
non-local irreducible components of the first resonance variety
$\mathcal{R}_1(\mathcal{A})$ are 2-dimensional and correspond to parallelograms $\mathcal{P}$ in
$\mathbb{C}^2=\mathbb{P}^2 \setminus H_{\infty}$ whose sides are in $\mathcal{A}$ and for
which $H_0$ is a diagonal.
Keywords:local system, line arrangement, characteristic variety, resonance variety Categories:14C21, 14F99, 32S22, 14E05, 14H50 |
28. CMB 2010 (vol 53 pp. 757)
Interval Pattern Avoidance for Arbitrary Root Systems
We extend the idea of interval pattern avoidance defined by Yong and
the author for $S_n$ to arbitrary Weyl groups using the definition of
pattern avoidance due to Billey and Braden, and Billey and Postnikov.
We show that, as previously shown by Yong and the
author for $\operatorname{GL}_n$, interval pattern avoidance is a universal tool for
characterizing which Schubert varieties have certain local properties,
and where these local properties hold.
Categories:14M15, 05E15 |
29. CMB 2010 (vol 53 pp. 746)
On Surfaces with p_{g}=0 and K^{2}=5 We construct new examples of surfaces of general type with $p_g=0$ and $K^2=5$ as ${\mathbb Z}_2 \times {\mathbb Z}_2$-covers and show that they are genus three hyperelliptic fibrations with bicanonical map of degree two.
Category:14J29 |
30. CMB 2009 (vol 53 pp. 218)
Restriction of the Tangent Bundle of $G/P$ to a Hypersurface Let P be a maximal proper parabolic subgroup of a connected simple linear algebraic group G, defined over $\mathbb C$, such that $n := \dim_{\mathbb C} G/P \geq 4$. Let $\iota \colon Z \hookrightarrow G/P$ be a reduced smooth hypersurface of degree at least $(n-1)\cdot \operatorname{degree}(T(G/P))/n$. We prove that the restriction of the tangent bundle $\iota^*TG/P$ is semistable.
Keywords:tangent bundle, homogeneous space, semistability, hypersurface Categories:14F05, 14J60, 14M15 |
31. CMB 2009 (vol 53 pp. 171)
Multiplicity-Free Schubert Calculus Multiplicity-free algebraic geometry is the study of subvarieties
$Y\subseteq X$ with the ``smallest invariants'' as witnessed by a
multiplicity-free Chow ring decomposition of
$[Y]\in A^{\star}(X)$ into a predetermined
linear basis.
This paper concerns the case of Richardson subvarieties of the Grassmannian
in terms of the Schubert basis. We give a nonrecursive combinatorial
classification of multiplicity-free Richardson varieties, i.e.,
we classify multiplicity-free products of Schubert classes. This answers
a question of W. Fulton.
Categories:14M15, 14M05, 05E99 |
32. CMB 2009 (vol 53 pp. 77)
Constructing (Almost) Rigid Rings and a UFD Having Infinitely Generated Derksen and Makar-Limanov Invariants |
Constructing (Almost) Rigid Rings and a UFD Having Infinitely Generated Derksen and Makar-Limanov Invariants An example is given of a UFD which has an infinitely generated Derksen invariant. The ring is "almost rigid" meaning that the Derksen invariant is equal to the Makar-Limanov invariant. Techniques to show that a ring is (almost) rigid are discussed, among which is a generalization of Mason's abc-theorem.
Categories:14R20, 13A50, 13N15 |
33. CMB 2009 (vol 53 pp. 247)
Root Extensions and Factorization in Affine Domains An integral domain R is IDPF (Irreducible Divisors of Powers Finite) if, for every non-zero element a in R, the ascending chain of non-associate irreducible divisors in R of $a^{n}$ stabilizes on a finite set as n ranges over the positive integers, while R is atomic if every non-zero element that is not a unit is a product of a finite number of irreducible elements (atoms). A ring extension S of R is a \emph{root extension} or \emph{radical extension} if for each s in S, there exists a natural number $n(s)$ with $s^{n(s)}$ in R. In this paper it is shown that the ascent and descent of the IDPF property and atomicity for the pair of integral domains $(R,S)$ is governed by the relative sizes of the unit groups $\operatorname{U}(R)$ and $\operatorname{U}(S)$ and whether S is a root extension of R. The following results are deduced from these considerations. An atomic IDPF domain containing a field of characteristic zero is completely integrally closed. An affine domain over a field of characteristic zero is IDPF if and only if it is completely integrally closed. Let R be a Noetherian domain with integral closure S. Suppose the conductor of S into R is non-zero. Then R is IDPF if and only if S is a root extension of R and $\operatorname{U}(S)/\operatorname{U}(R)$ is finite.
Categories:13F15, 14A25 |
34. CMB 2009 (vol 53 pp. 58)
Ranks in Families of Jacobian Varieties of Twisted Fermat Curves In this paper, we prove that the unboundedness of ranks in families of Jacobian varieties of twisted Fermat curves is equivalent to the divergence of certain infinite series.
Keywords:Fermat curve, Jacobian variety, elliptic curve, canonical height Categories:11G10, 11G05, 11G50, 14G05, 11G30, 14H45, 14K15 |
35. CMB 2009 (vol 52 pp. 535)
A Note on Locally Nilpotent Derivations\\ and Variables of $k[X,Y,Z]$ We strengthen certain results
concerning actions of $(\Comp,+)$ on $\Comp^{3}$
and embeddings of $\Comp^{2}$ in $\Comp^{3}$,
and show that these results are in fact valid
over any field of characteristic zero.
Keywords:locally nilpotent derivations, group actions, polynomial automorphisms, variable, affine space Categories:14R10, 14R20, 14R25, 13N15 |
36. CMB 2009 (vol 52 pp. 493)
A One-Dimensional Family of $K3$ Surfaces with a $\Z_4$ Action The minimal resolution of the degree four cyclic cover of the plane
branched along a GIT stable quartic is a $K3$ surface with a non
symplectic action of $\Z_4$. In this paper
we study the geometry of the one-dimensional family of $K3$ surfaces
associated to the locus of plane quartics with five nodes.
Keywords:genus three curves, $K3$ surfaces Categories:14J28, 14J50, 14J10 |
37. CMB 2009 (vol 52 pp. 175)
Connections on a Parabolic Principal Bundle, II In \emph{Connections on a parabolic principal bundle over a curve, I}
we defined connections on a parabolic
principal bundle. While connections on usual principal bundles are
defined as splittings of the Atiyah exact sequence, it was noted in
the above article that the Atiyah exact sequence does not generalize to
the parabolic principal bundles.
Here we show that a twisted version
of the Atiyah exact sequence generalizes to the context of
parabolic principal bundles. For usual principal bundles, giving a
splitting of this twisted Atiyah exact sequence is equivalent
to giving a splitting of the Atiyah exact sequence. Connections on
a parabolic principal bundle can be defined using the
generalization of the twisted Atiyah exact sequence.
Keywords:Parabolic bundle, Atiyah exact sequence, connection Categories:32L05, 14F05 |
38. CMB 2009 (vol 52 pp. 224)
Equations and Complexity for the Dubois--Efroymson Dimension Theorem Let $\R$ be a real closed field, let $X \subset \R^n$ be an
irreducible real algebraic set and let $Z$ be an algebraic subset of
$X$ of codimension $\geq 2$. Dubois and Efroymson proved the existence
of an irreducible algebraic subset of $X$ of codimension $1$
containing~$Z$. We improve this dimension theorem as follows. Indicate
by $\mu$ the minimum integer such that the ideal of polynomials in
$\R[x_1,\ldots,x_n]$ vanishing on $Z$ can be generated by polynomials
of degree $\leq \mu$. We prove the following two results:
\begin{inparaenum}[\rm(1)]
\item There
exists a polynomial $P \in \R[x_1,\ldots,x_n]$ of degree~$\leq \mu+1$
such that $X \cap P^{-1}(0)$ is an irreducible algebraic subset of $X$
of codimension $1$ containing~$Z$.
\item Let $F$ be a polynomial in
$\R[x_1,\ldots,x_n]$ of degree~$d$ vanishing on $Z$. Suppose there
exists a nonsingular point $x$ of $X$ such that $F(x)=0$ and the
differential at $x$ of the restriction of $F$ to $X$ is nonzero. Then
there exists a polynomial $G \in \R[x_1,\ldots,x_n]$ of degree $\leq
\max\{d,\mu+1\}$ such that, for each $t \in (-1,1) \setminus \{0\}$,
the set $\{x \in X \mid F(x)+tG(x)=0\}$ is an irreducible algebraic
subset of $X$ of codimension $1$ containing~$Z$.
\end{inparaenum} Result (1) and a
slightly different version of result~(2) are valid over any
algebraically closed field also.
Keywords:Irreducible algebraic subvarieties, complexity of algebraic varieties, Bertini's theorems Categories:14P05, 14P20 |
39. CMB 2009 (vol 52 pp. 200)
Schubert Calculus on a Grassmann Algebra The ({\em classical}, {\em small quantum}, {\em equivariant})
cohomology ring of the grassmannian $G(k,n)$ is generated by
certain derivations operating on an exterior algebra of a free
module of rank $n$ ( Schubert calculus on a Grassmann
algebra). Our main result gives, in a unified way, a presentation
of all such cohomology rings in terms of generators and
relations. Using results of Laksov and Thorup, it also provides
a presentation of the universal
factorization algebra of a monic polynomial of degree $n$ into the
product of two monic polynomials, one of degree $k$.
Categories:14N15, 14M15 |
40. CMB 2009 (vol 52 pp. 161)
A New Tautological Relation in $\overline{\mathcal{M}}_{3,1}$ via the Invariance Constraint A new tautological relation of $\overline{\mathcal{M}}_{3,1}$ in codimension 3
is derived and proved, using an invariance constraint from
our previous work.
Category:14H10 |
41. CMB 2009 (vol 52 pp. 117)
On the Rational Points of the Curve $f(X,Y)^q = h(X)g(X,Y)$ Let $q = 2,3$ and $f(X,Y)$, $g(X,Y)$, $h(X)$ be polynomials with
integer coefficients. In this paper we deal with the curve
$f(X,Y)^q = h(X)g(X,Y)$, and we show that under some favourable
conditions it is possible to determine all of its rational points.
Categories:11G30, 14G05, 14G25 |
42. CMB 2009 (vol 52 pp. 39)
A Representation Theorem for Archimedean Quadratic Modules on $*$-Rings We present a new approach to noncommutative real algebraic geometry
based on the representation theory of $C^\ast$-algebras.
An important result in commutative real algebraic geometry is
Jacobi's representation theorem for archimedean quadratic modules
on commutative rings.
We show that this theorem is a consequence of the
Gelfand--Naimark representation theorem for commutative $C^\ast$-algebras.
A noncommutative version of Gelfand--Naimark theory was studied by
I. Fujimoto. We use his results to generalize
Jacobi's theorem to associative rings with involution.
Keywords:Ordered rings with involution, $C^\ast$-algebras and their representations, noncommutative convexity theory, real algebraic geometry Categories:16W80, 46L05, 46L89, 14P99 |
43. CMB 2008 (vol 51 pp. 519)
The Effective Cone of the Kontsevich Moduli Space In this paper we prove that the cone of effective divisors on the
Kontsevich moduli spaces of stable maps, $\Kgnb{0,0}(\PP^r,d)$,
stabilize when $r \geq d$. We give a complete characterization of the
effective divisors on $\Kgnb{0,0}(\PP^d,d)$. They are non-negative
linear combinations of boundary divisors and the divisor of maps with
degenerate image.
Categories:14D20, 14E99, 14H10 |
44. CMB 2008 (vol 51 pp. 283)
The Noether--Lefschetz Theorem Via Vanishing of Coherent Cohomology We prove that for a generic hypersurface in $\mathbb P^{2n+1}$ of degree at
least $2+2/n$, the $n$-th Picard number is one. The proof is algebraic
in nature and follows from certain coherent cohomology vanishing.
Keywords:Noether--Lefschetz, algebraic cycles, Picard number Categories:14C15, 14C25 |
45. CMB 2008 (vol 51 pp. 125)
Explicit Real Cubic Surfaces The topological classification of smooth real
cubic surfaces is
recalled and compared to the classification in terms of
the number of real lines and of real tritangent planes,
as obtained
by L.~Schl\"afli in 1858.
Using this, explicit examples of
surfaces of every possible type are given.
Categories:14J25, 14J80, 14P25, 14Q10 |
46. CMB 2008 (vol 51 pp. 114)
Zero Cycles on a Twisted Cayley Plane Let $k$ be a field of characteristic not $2,3$.
Let $G$ be an exceptional simple algebraic group over $k$
of type $\F$, $^1{\E_6}$ or $\E_7$ with trivial Tits algebras.
Let $X$ be a projective $G$-homogeneous variety.
If $G$ is of type $\E_7$, we assume in addition
that the respective
parabolic subgroup is of type $P_7$.
The main result of the paper says that
the degree map on the group of zero cycles of $X$
is injective.
Categories:20G15, 14C15 |
47. CMB 2007 (vol 50 pp. 486)
Higher-Dimensional Modular\\Calabi--Yau Manifolds We construct several examples of higher-dimensional Calabi--Yau manifolds and prove their
modularity.
Categories:14G10, 14J32, 11G40 |
48. CMB 2007 (vol 50 pp. 567)
Exotic Torsion, Frobenius Splitting and the Slope Spectral Sequence In this paper we show that any Frobenius split, smooth, projective
threefold over a perfect field of characteristic $p>0$ is
Hodge--Witt. This is proved by generalizing to the case of
threefolds a well-known criterion due to N.~Nygaard for surfaces to be Hodge-Witt.
We also show that the second crystalline
cohomology of any smooth, projective Frobenius split variety does
not have any exotic torsion. In the last two sections we include
some applications.
Keywords:threefolds, Frobenius splitting, Hodge--Witt, crystalline cohomology, slope spectral sequence, exotic torsion Categories:14F30, 14J30 |
49. CMB 2007 (vol 50 pp. 427)
On the Image of Certain Extension Maps.~I Let $X$ be a smooth complex projective curve of genus $g\geq
1$. Let $\xi\in J^1(X)$ be a line bundle on $X$ of degree $1$. Let
$W=\Ext^1(\xi^n,\xi^{-1})$ be the space of extensions of $\xi^n$
by $\xi^{-1}$. There is a rational map
$D_{\xi}\colon G(n,W)\rightarrow SU_{X}(n+1)$,
where $G(n,W)$ is the Grassmannian variety of $n$-linear subspaces
of $W$ and $\SU_{X}(n+1)$ is the moduli space of rank $n+1$ semi-stable
vector
bundles on $X$ with trivial determinant. We prove that if $n=2$,
then $D_{\xi}$ is
everywhere defined and is injective.
Categories:14H60, 14F05, 14D20 |
50. CMB 2007 (vol 50 pp. 196)
Plane Quartic Twists of $X(5,3)$ Given an odd surjective Galois representation $\varrho\from \G_\Q\to\PGL_2(\F_3)$ and a
positive integer~$N$, there exists a twisted modular curve $X(N,3)_\varrho$
defined over $\Q$ whose rational points classify the quadratic $\Q$-curves of degree $N$
realizing~$\varrho$. This paper gives a method to provide an explicit plane quartic model for
this curve in the genus-three case $N=5$.
Categories:11F03, 11F80, 14G05 |