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1. CMB Online first

Kaveh, Kiumars; Khovanskii, A. G.
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

2. CMB Online first

Parusiński, Adam; Rolin, Jean-Philippe
Note on the Weierstrass 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

3. CMB 2013 (vol 57 pp. 439)

Yang, YanHong
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

4. CMB 2012 (vol 57 pp. 303)

Gille, Philippe
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

5. CMB 2012 (vol 57 pp. 97)

Levy, Jason
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

6. 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

7. 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, Shafarevich-Tate group
Categories:11G35, 14G25

8. CMB 2011 (vol 56 pp. 500)

Browning, T. D.
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

9. CMB 2011 (vol 55 pp. 842)

Sairaiji, Fumio; Yamauchi, Takuya
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

10. CMB 2011 (vol 55 pp. 752)

Hickel, M.; Rond, G.
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 Płoski smoothing theorem in this case.

Keywords:Artin approximation, real analytic equations
Categories:13B40, 13L05, 14F12

11. 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

12. 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

13. CMB 2011 (vol 55 pp. 319)

Jardine, J. F.
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

14. 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, Eisenstein-Kronecker series, $L$-series of $K3$-surfaces, $l$-adic representations, Livné criterion, Rankin-Cohen brackets
Categories:11, 14D, 14J

15. CMB 2011 (vol 54 pp. 430)

DeLand, Matthew
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

16. 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, Kähler 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

17. 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 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

18. 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

19. CMB 2010 (vol 54 pp. 381)

Velušček, Dejan
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

20. 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 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

21. 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

22. CMB 2010 (vol 53 pp. 746)

Werner, Caryn
On Surfaces with pg=0 and K2=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

23. 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 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

24. CMB 2009 (vol 53 pp. 171)

Thomas, Hugh; Yong, Alexander
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

25. CMB 2009 (vol 53 pp. 77)

Finston, David; Maubach, Stefan
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
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