26. CMB 2014 (vol 58 pp. 80)
 Harada, Megumi; Horiguchi, Tatsuya; Masuda, Mikiya

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 FukukawaHaradaMasuda,
which was only for Lie type $A$.
Keywords:equivariant cohomology, Peterson varieties, flag varieties, Monk formula, Giambelli formula Categories:55N91, 14N15 

27. CMB 2014 (vol 58 pp. 356)
 Sebag, Julien

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 

28. CMB 2014 (vol 57 pp. 749)
 Cavalieri, Renzo; Marcus, Steffen

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 

29. CMB 2014 (vol 57 pp. 658)
 Thang, Nguyen Tat

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 

30. CMB 2013 (vol 57 pp. 562)
 Kaveh, Kiumars; Khovanskii, A. G.

Note on the Grothendieck Group of Subspaces of Rational Functions and Shokurov's Cartier bdivisors
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 bdivisor, Grothendieck group Categories:14C20, 14Exx 

31. CMB 2013 (vol 57 pp. 614)
32. CMB 2013 (vol 57 pp. 439)
 Yang, YanHong

The Fixed Point Locus of the Verschiebung on $\mathcal{M}_X(2, 0)$ for Genus2 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 

33. CMB 2012 (vol 57 pp. 97)
 Levy, Jason

Rationality and the JordanGattiViniberghi decomposition
We verify
our earlier conjecture
and use it to prove that the
semisimple parts of the rational JordanKacVinberg 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 

34. CMB 2012 (vol 57 pp. 303)
35. CMB 2012 (vol 56 pp. 640)
 Türkmen, İnan Utku

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 

36. CMB 2011 (vol 56 pp. 225)
 Agashe, Amod

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, ShafarevichTate group Categories:11G35, 14G25 

37. CMB 2011 (vol 56 pp. 500)
 Browning, T. D.

The LangWeil 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 

38. CMB 2011 (vol 55 pp. 842)
39. CMB 2011 (vol 55 pp. 752)
40. CMB 2011 (vol 55 pp. 850)
 Shparlinski, Igor E.; Stange, Katherine E.

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 

41. CMB 2011 (vol 55 pp. 799)
 Novelli, Carla; Occhetta, Gianluca

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 

42. CMB 2011 (vol 55 pp. 319)
 Jardine, J. F.

The Verdier Hypercovering Theorem
This note gives a simple cocycletheoretic 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 

43. CMB 2011 (vol 55 pp. 26)
 Bertin, Marie José

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, EisensteinKronecker series, $L$series of $K3$surfaces, $l$adic representations, LivnÃ© criterion, RankinCohen brackets Categories:11, 14D, 14J 

44. CMB 2011 (vol 54 pp. 430)
 DeLand, Matthew

Complete Families of Linearly Nondegenerate Rational Curves
We prove that every complete family of linearly nondegenerate
rational curves of degree $e > 2$ in $\mathbb{P}^{n}$ has at most $n1$
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 

45. CMB 2011 (vol 54 pp. 472)
 Iacono, Donatella

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 

46. 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 

47. 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 

48. 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 

49. 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 

50. 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 
