1. CMB Online first
 Reichstein, Zinovy; Vistoli, Angelo

On the dimension of the locus of determinantal hypersurfaces
The characteristic polynomial $P_A(x_0, \dots,
x_r)$
of an $r$tuple $A := (A_1, \dots, A_r)$ of $n \times n$matrices
is
defined as
\[ P_A(x_0, \dots, x_r) := \det(x_0 I + x_1 A_1 + \dots + x_r
A_r) \, . \]
We show that if $r \geqslant 3$
and $A := (A_1, \dots, A_r)$ is an $r$tuple of $n \times n$matrices in general position,
then up to conjugacy, there are only finitely many $r$tuples
$A' := (A_1', \dots, A_r')$ such that $p_A = p_{A'}$. Equivalently,
the locus of determinantal hypersurfaces of degree $n$ in $\mathbf{P}^r$
is irreducible of dimension $(r1)n^2 + 1$.
Keywords:determinantal hypersurface, matrix invariant, $q$binomial coefficient Categories:14M12, 15A22, 05A10 

2. CMB 2016 (vol 59 pp. 865)
 Pal, Sarbeswar

Moduli of Rank 2 Stable Bundles and Hecke Curves
Let $X$ be smooth projective curve of arbitrary genus $g \gt 3$
over complex numbers. In this short note we will show that the
moduli
space of rank $2$ stable vector bundles with determinant isomorphic
to $L_x$, where $L_x$ denote the line bundle corresponding to
a point $x \in X$ is isomorphic to certain lines in the moduli
space of Sequivalence classes of semistable bundles of rank
2 with
trivial determinant.
Keywords:Hecke curve, (0,1) stable bundle Category:14D21 

3. CMB Online first
 Iena, Oleksandr; Leytem, Alain

On the singular sheaves in the fine Simpson moduli spaces of $1$dimensional sheaves
In the Simpson moduli space $M$ of semistable sheaves with
Hilbert polynomial $dm1$ on a projective plane we study the
closed subvariety $M'$ of sheaves that are not locally free on
their support. We show that for $d\ge 4$ it is a singular subvariety
of codimension $2$ in $M$. The blow up of $M$ along $M'$ is interpreted
as a (partial) modification of $M\setminus M'$ by line bundles
(on support).
Keywords:Simpson moduli spaces, coherent sheaves, vector bundles on curves, singular sheaves Category:14D20 

4. CMB Online first
5. CMB Online first
 Haase, Christian; Hofmann, Jan

Convexnormal (pairs of) polytopes
In 2012 Gubeladze (Adv. Math. 2012)
introduced the notion of $k$convexnormal polytopes to show
that
integral polytopes all of whose edges are longer than $4d(d+1)$
have
the integer decomposition property.
In the first part of this paper we show that for lattice polytopes
there is no difference between $k$ and $(k+1)$convexnormality
(for
$k\geq 3 $) and improve the bound to $2d(d+1)$. In the second
part we
extend the definition to pairs of polytopes. Given two rational
polytopes $P$ and $Q$, where the normal fan of $P$ is a refinement
of
the normal fan of $Q$.
If every edge $e_P$ of $P$ is at least $d$ times as long as the
corresponding face (edge or vertex) $e_Q$ of $Q$, then $(P+Q)\cap
\mathbb{Z}^d
= (P\cap \mathbb{Z}^d ) + (Q \cap \mathbb{Z}^d)$.
Keywords:integer decomposition property, integrally closed, projectively normal, lattice polytopes Categories:52B20, 14M25, 90C10 

6. CMB 2016 (vol 59 pp. 760)
 Fichou, Goulwen; Quarez, Ronan; Shiota, Masahiro

Artin Approximation Compatible with a Change of Variables
We propose a version of the classical Artin
approximation
which allows to perturb the variables of the approximated solution. Namely, it is possible to approximate a formal solution of a
Nash equation by a Nash solution in a
compatible way with a given Nash change of variables.
This result is closely related to the socalled nested Artin
approximation and becomes false in the analytic setting. We provide
local and global versions of this approximation in real and complex
geometry together with an application to the RightLeft equivalence
of Nash maps.
Keywords:Artin approximation, global case, Nash functions Categories:14P20, 58A07 

7. CMB 2016 (vol 59 pp. 824)
 Karpenko, Nikita A.

Incompressibility of Products of Pseudohomogeneous Varieties
We show that the conjectural criterion of $p$incompressibility
for products of projective homogeneous varieties in terms of
the factors, previously known in a few special cases only, holds
in general.
Actually, the proof goes through for a wider class of varieties
which includes the norm varieties associated to symbols in Galois
cohomology of arbitrary degree.
Keywords:algebraic groups, projective homogeneous varieties, Chow groups and motives, canonical dimension and incompressibility Categories:20G15, 14C25 

8. CMB 2016 (vol 59 pp. 449)
 Abdallah, Nancy

On Hodge Theory of Singular Plane Curves
The dimensions of the graded quotients of the
cohomology of a plane curve complement $U=\mathbb P^2 \setminus C$
with respect to the Hodge filtration are described in terms of
simple geometrical invariants. The case of curves with ordinary
singularities is discussed in detail. We also give a precise
numerical estimate for the difference between the Hodge filtration
and the pole order filtration on $H^2(U,\mathbb C)$.
Keywords:plane curves, Hodge and pole order filtrations Categories:32S35, 32S22, 14H50 

9. CMB 2016 (vol 59 pp. 858)
 Osserman, Brian

Stability of Vector Bundles on Curves and Degenerations
We introduce a weaker notion of (semi)stability for vector bundles
on
reducible curves which does not depend on a choice of polarization,
and
which suffices for many applications of degeneration techniques.
We explore the basic
properties of this alternate notion of (semi)stability. In a
complementary
direction, we record a proof of the existence of semistable extensions
of vector bundles in suitable degenerations.
Keywords:vector bundle, stability, degeneration Categories:14D06, 14H60 

10. CMB 2016 (vol 59 pp. 311)
 Ilten, Nathan; Teitler, Zach

Product Ranks of the $3\times 3$ Determinant and Permanent
We show that the product rank of the $3 \times 3$ determinant
$\det_3$ is $5$,
and the product rank of the $3 \times 3$ permanent
$\operatorname{perm}_3$
is $4$.
As a corollary, we obtain that the tensor rank of $\det_3$ is
$5$ and the tensor rank of $\operatorname{perm}_3$ is $4$.
We show moreover that the border product rank of $\operatorname{perm}_n$ is
larger than $n$ for any $n\geq 3$.
Keywords:product rank, tensor rank, determinant, permanent, Fano schemes Categories:15A21, 15A69, 14M12, 14N15 

11. CMB 2015 (vol 58 pp. 673)
 Achter, Jeffrey; Williams, Cassandra

Local Heuristics and an Exact Formula for Abelian Surfaces Over Finite Fields
Consider a quartic $q$Weil polynomial $f$. Motivated by equidistribution
considerations, we define, for each prime $\ell$, a local factor
that
measures the relative frequency with which $f\bmod \ell$ occurs
as the
characteristic polynomial of a symplectic similitude over $\mathbb{F}_\ell$.
For a certain
class of polynomials, we show that the resulting infinite product
calculates the number of principally polarized abelian surfaces
over $\mathbb{F}_q$
with Weil polynomial $f$.
Keywords:abelian surfaces, finite fields, random matrices Category:14K02 

12. CMB 2015 (vol 59 pp. 144)
 Laterveer, Robert

A Brief Note Concerning Hard Lefschetz for Chow Groups
We formulate a conjectural hard Lefschetz property
for Chow groups, and prove this in some special cases: roughly
speaking, for varieties with finitedimensional motive, and
for varieties whose selfproduct has vanishing middledimensional
Griffiths group. An appendix includes related statements that
follow from results of Vial.
Keywords:algebraic cycles, Chow groups, finitedimensional motives Categories:14C15, 14C25, 14C30 

13. CMB 2015 (vol 58 pp. 620)
 Sands, Jonathan W.

$L$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 nontrivial 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 $s=1n$.
Keywords:motivic cohomology, regulator, Artin Lfunctions Categories:11R42, 11R70, 14F42, 19F27 

14. CMB 2015 (vol 58 pp. 519)
 Kang, SuJeong

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 

15. CMB 2015 (vol 58 pp. 250)
 Cartwright, Dustin; Jensen, David; Payne, Sam

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, BrillNoether theory, special divisors on algebraic curves Categories:14T05, 14H51 

16. 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 

17. 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 

18. 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 

19. 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 

20. 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 

21. CMB 2013 (vol 57 pp. 614)
22. 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 

23. 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 

24. CMB 2012 (vol 57 pp. 303)
25. 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 
