26. CMB 2010 (vol 54 pp. 561)
 Uren, James J.

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 socalled \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 

27. CMB 2010 (vol 54 pp. 520)
 Polishchuk, A.

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 

28. CMB 2010 (vol 54 pp. 381)
 Velušček, Dejan

A Short Note on the Higher Level Version of the KrullBaer Theorem
Klep and Velu\v{s}\v{c}ek generalized the KrullBaer theorem for
higher level preorderings to the noncommutative 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 KrullBaer theorem for preorderings.
Keywords:orderings of higher level, division rings, valuations Categories:14P99, 06Fxx 

29. CMB 2010 (vol 54 pp. 56)
 Dinh, Thi Anh Thu

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
nonlocal irreducible components of the first resonance variety
$\mathcal{R}_1(\mathcal{A})$ are 2dimensional 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 

30. CMB 2010 (vol 53 pp. 757)
 Woo, Alexander

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 

31. CMB 2010 (vol 53 pp. 746)
 Werner, Caryn

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 

32. CMB 2009 (vol 53 pp. 247)
 Etingof, P.; Malcolmson, P.; Okoh, F.

Root Extensions and Factorization in Affine Domains
An integral domain R is IDPF (Irreducible Divisors of Powers Finite) if, for every nonzero element a in R, the ascending chain of nonassociate 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 nonzero 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 nonzero. 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 

33. CMB 2009 (vol 53 pp. 218)
 Biswas, Indranil

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 $(n1)\cdot \operatorname{degree}(T(G/P))/n$. 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 

34. CMB 2009 (vol 53 pp. 58)
 Dąbrowski, Andrzej; Jędrzejak, Tomasz

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 53 pp. 171)
 Thomas, Hugh; Yong, Alexander

MultiplicityFree Schubert Calculus
Multiplicityfree algebraic geometry is the study of subvarieties
$Y\subseteq X$ with the ``smallest invariants'' as witnessed by a
multiplicityfree 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 multiplicityfree Richardson varieties, i.e.,
we classify multiplicityfree products of Schubert classes. This answers
a question of W. Fulton.
Categories:14M15, 14M05, 05E99 

36. CMB 2009 (vol 53 pp. 77)
37. CMB 2009 (vol 52 pp. 535)
 Daigle, Daniel; Kaliman, Shulim

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 

38. CMB 2009 (vol 52 pp. 493)
 Artebani, Michela

A OneDimensional 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 onedimensional family of $K3$ surfaces
associated to the locus of plane quartics with five nodes.
Keywords:genus three curves, $K3$ surfaces Categories:14J28, 14J50, 14J10 

39. CMB 2009 (vol 52 pp. 175)
 Biswas, Indranil

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 

40. CMB 2009 (vol 52 pp. 224)
 Ghiloni, Riccardo

Equations and Complexity for the DuboisEfroymson 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 

41. CMB 2009 (vol 52 pp. 200)
 Gatto, Letterio; Santiago, Ta\'\i se

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 

42. CMB 2009 (vol 52 pp. 161)
43. CMB 2009 (vol 52 pp. 117)
 Poulakis, Dimitrios

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 

44. CMB 2009 (vol 52 pp. 39)
 Cimpri\v{c}, Jakob

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
GelfandNaimark representation theorem for commutative $C^\ast$algebras.
A noncommutative version of GelfandNaimark 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 

45. CMB 2008 (vol 51 pp. 519)
 Coskun, Izzet; Harris, Joe; Starr, Jason

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 nonnegative
linear combinations of boundary divisors and the divisor of maps with
degenerate image.
Categories:14D20, 14E99, 14H10 

46. CMB 2008 (vol 51 pp. 283)
47. CMB 2008 (vol 51 pp. 125)
 PoloBlanco, Irene; Top, Jaap

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 

48. CMB 2008 (vol 51 pp. 114)
 Petrov, V.; Semenov, N.; Zainoulline, K.

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 

49. CMB 2007 (vol 50 pp. 486)
50. CMB 2007 (vol 50 pp. 567)
 Joshi, Kirti

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
HodgeWitt. This is proved by generalizing to the case of
threefolds a wellknown criterion due to N.~Nygaard for surfaces to be HodgeWitt.
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, HodgeWitt, crystalline cohomology, slope spectral sequence, exotic torsion Categories:14F30, 14J30 
