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1. CMB Online first

Kocel-Cynk, Beata; Pawłucki, Wiesław; Valette, Anna
$\mathcal{C}^p$-parametrization in o-minimal structures
We give a geometric and elementary proof of the uniform $\mathcal C^p$-parametrization theorem of Yomdin-Gromov in arbitrary o-minimal structures.

Keywords:o-minimal structure, $C^p$-parametrization
Categories:03C64, 14P15, 32B20

2. CMB Online first

González, José Luis; Karu, Kalle
Examples of non-finitely generated Cox rings
We bring examples of toric varieties blown up at a point in the torus that do not have finitely generated Cox rings. These examples are generalizations of our earlier work, where toric surfaces of Picard number $1$ were studied. In this article we consider toric varieties of higher Picard number and higher dimension. In particular, we bring examples of weighted projective $3$-spaces blown up at a point that do not have finitely generated Cox rings.

Keywords:Cox ring, Mori dream space, toric variety
Categories:14M25, 14C20, 14E30

3. CMB Online first

Purbhoo, Kevin
Total nonnegativity and stable polynomials
We consider homogeneous multiaffine polynomials whose coefficients are the Plücker coordinates of a point $V$ of the Grassmannian. We show that such a polynomial is stable (with respect to the upper half plane) if and only if $V$ is in the totally nonnegative part of the Grassmannian. To prove this, we consider an action of matrices on multiaffine polynomials. We show that a matrix $A$ preserves stability of polynomials if and only if $A$ is totally nonnegative. The proofs are applications of classical theory of totally nonnegative matrices, and the generalized Pólya-Schur theory of Borcea and Brändén.

Keywords:stable polynomial, zeros of a complex polynomial, total nonnegative Grassmannian, totally nonnegative matrix
Categories:32A60, 14M15, 14P10, 15B48

4. CMB Online first

Asgarli, Shamil
Sharp Bertini theorem for plane curves over finite fields
We prove that if $C$ is a reflexive smooth plane curve of degree $d$ defined over a finite field $\mathbb{F}_q$ with $d\leq q+1$, then there is an $\mathbb{F}_q$-line $L$ that intersects $C$ transversely. We also prove the same result for non-reflexive curves of degree $p+1$ and $2p+1$ where $q=p^{r}$.

Keywords:Bertini theorem, transversality, finite field
Categories:14H50, 11G20, 14N05

5. CMB Online first

Sun, Chia-Liang
Weak approximation for points with coordinates in rank-one subgroups of global function fields
For every affine variety over a global function field, we show that the set of its points with coordinates in an arbitrary rank-one multiplicative subgroup of this function field satisfies the required property of weak approximation for finite sets of places of this function field avoiding arbitrarily given finitely many places.

Keywords:weak approximation, global function fields, local-global criteria
Categories:14G05, 11R58

6. CMB Online first

Bertapelle, A.; Mazzari, N.
On deformations of $1$-motives
According to a well-known theorem of Serre and Tate, the infinitesimal deformation theory of an abelian variety in positive characteristic is equivalent to the infinitesimal deformation theory of its Barsotti-Tate group. We extend this result to $1$-motives.

Keywords:$1$-motive, Barsotti-Tate group
Categories:14L15, 14C15, 14L05

7. CMB Online first

Zhang, Zheng
On motivic realizations of the canonical Hermitian variations of Hodge structure of Calabi-Yau type over type $D^{\mathbb H}$ domains
Let $\mathcal{D}$ be the irreducible Hermitian symmetric domain of type $D_{2n}^{\mathbb{H}}$. There exists a canonical Hermitian variation of real Hodge structure $\mathcal{V}_{\mathbb{R}}$ of Calabi-Yau type over $\mathcal{D}$. This short note concerns the problem of giving motivic realizations for $\mathcal{V}_{\mathbb{R}}$. Namely, we specify a descent of $\mathcal{V}_{\mathbb{R}}$ from $\mathbb{R}$ to $\mathbb{Q}$ and ask whether the $\mathbb{Q}$-descent of $\mathcal{V}_{\mathbb{R}}$ can be realized as sub-variation of rational Hodge structure of those coming from families of algebraic varieties. When $n=2$, we give a motivic realization for $\mathcal{V}_{\mathbb{R}}$. When $n \geq 3$, we show that the unique irreducible factor of Calabi-Yau type in $\mathrm{Sym}^2 \mathcal{V}_{\mathbb{R}}$ can be realized motivically.

Keywords:variations of Hodge structure, Hermitian symmetric domain
Categories:14D07, 32G20, 32M15

8. CMB 2017 (vol 61 pp. 659)

Wang, Zhenjian
On Deformations of Nodal Hypersurfaces
We extend the infinitesimal Torelli theorem for smooth hypersurfaces to nodal hypersurfaces.

Keywords:nodal hypersurface, deformation, Torelli theorem
Categories:32S35, 14C30, 14D07, 32S25

9. CMB 2017 (vol 61 pp. 650)

Shirane, Taketo
Connected numbers and the embedded topology of plane curves
The splitting number of a plane irreducible curve for a Galois cover is effective to distinguish the embedded topology of plane curves. In this paper, we define the connected number of a plane curve (possibly reducible) for a Galois cover, which is similar to the splitting number. By using the connected number, we distinguish the embedded topology of Artal arrangements of degree $b\geq 4$, where an Artal arrangement of degree $b$ is a plane curve consisting of one smooth curve of degree $b$ and three of its total inflectional tangen

Keywords:plane curve, splitting curve, Zariski pair, cyclic cover, splitting number
Categories:14H30, 14H50, 14F45

10. CMB Online first

Reichstein, Zinovy B.
On a property of real plane curves of even degree
F. Cukierman asked whether or not for every smooth real plane curve $X \subset \mathbb{P}^2$ of even degree $d \geqslant 2$ there exists a real line $L \subset \mathbb{P}^2$ such $X \cap L$ has no real points. We show that the answer is ``yes" if $d = 2$ or $4$ and ``no" if $n \geqslant 6$.

Keywords:real algebraic geometry, plane curve, maximizer function, bitangent
Categories:14P05, 14H50

11. CMB 2017 (vol 61 pp. 572)

Koskivirta, Jean-Stefan
Normalization of closed Ekedahl-Oort strata
We apply our theory of partial flag spaces developed with W. Goldring to study a group-theoretical generalization of the canonical filtration of a truncated Barsotti-Tate group of level 1. As an application, we determine explicitly the normalization of the Zariski closures of Ekedahl-Oort strata of Shimura varieties of Hodge-type as certain closed coarse strata in the associated partial flag spaces.

Keywords:Ekedahl-Oort stratification, Shimura variety
Categories:14K10, 20G40, 11G18

12. CMB 2017 (vol 61 pp. 608)

Loeffler, David
A note on $p$-adic Rankin-Selberg $L$-functions
We prove an interpolation formula for the values of certain $p$-adic Rankin-Selberg $L$-functions associated to non-ordinary modular forms.

Keywords:$p$-adic $L$-function, Iwasawa theory
Categories:11F85, 11F67, 11G40, 14G35

13. CMB 2017 (vol 61 pp. 328)

Maican, Mario
Moduli of Space Sheaves with Hilbert Polynomial $4m+1$
We investigate the moduli space of sheaves supported on space curves of degree $4$ and having Euler characteristic $1$. We give an elementary proof of the fact that this moduli space consists of three irreducible components.

Keywords:moduli of sheaves, semi-stable sheaves
Categories:14D20, 14D22

14. CMB 2017 (vol 61 pp. 272)

Franz, Matthias
Symmetric Products of Equivariantly Formal Spaces
Let \(X\) be a CW complex with a continuous action of a topological group \(G\). We show that if \(X\) is equivariantly formal for singular cohomology with coefficients in some field \(\Bbbk\), then so are all symmetric products of \(X\) and in fact all its \(\Gamma\)-products. In particular, symmetric products of quasi-projective M-varieties are again M-varieties. This generalizes a result by Biswas and D'Mello about symmetric products of M-curves. We also discuss several related questions.

Keywords:symmetric product, equivariant formality, maximal variety, Gamma product
Categories:55N91, 55S15, 14P25

15. CMB 2017 (vol 60 pp. 490)

Fiori, Andrew
A Riemann-Hurwitz Theorem for the Algebraic Euler Characteristic
We prove an analogue of the Riemann-Hurwitz theorem for computing Euler characteristics of pullbacks of coherent sheaves through finite maps of smooth projective varieties in arbitrary dimensions, subject only to the condition that the irreducible components of the branch and ramification locus have simple normal crossings.

Keywords:Riemann-Hurwitz, logarithmic-Chern class, Euler characteristic
Categories:14F05, 14C17

16. CMB 2017 (vol 61 pp. 201)

Takahashi, Tomokuni
Projective plane bundles over an elliptic curve
We calculate the dimension of cohomology groups for the holomorphic tangent bundles of each isomorphism class of the projective plane bundle over an elliptic curve. As an application, we construct the families of projective plane bundles, and prove that the families are effectively parametrized and complete.

Keywords:projective plane bundle, vector bundle, elliptic curve, deformation, Kodaira-Spencer map
Categories:14J10, 14J30, 14D15

17. CMB 2017 (vol 60 pp. 478)

Carrell, Jim; Kaveh, Kiumars
Springer's Weyl Group Representation via Localization
Let $G$ denote a reductive algebraic group over $\mathbb{C}$ and $x$ a nilpotent element of its Lie algebra $\mathfrak{g}$. The Springer variety $\mathcal{B}_x$ is the closed subvariety of the flag variety $\mathcal{B}$ of $G$ parameterizing the Borel subalgebras of $\mathfrak{g}$ containing $x$. It has the remarkable property that the Weyl group $W$ of $G$ admits a representation on the cohomology of $\mathcal{B}_x$ even though $W$ rarely acts on $\mathcal{B}_x$ itself. Well-known constructions of this action due to Springer et al use technical machinery from algebraic geometry. The purpose of this note is to describe an elementary approach that gives this action when $x$ is what we call parabolic-surjective. The idea is to use localization to construct an action of $W$ on the equivariant cohomology algebra $H_S^*(\mathcal{B}_x)$, where $S$ is a certain algebraic subtorus of $G$. This action descends to $H^*(\mathcal{B}_x)$ via the forgetful map and gives the desired representation. The parabolic-surjective case includes all nilpotents of type $A$ and, more generally, all nilpotents for which it is known that $W$ acts on $H_S^*(\mathcal{B}_x)$ for some torus $S$. Our result is deduced from a general theorem describing when a group action on the cohomology of the fixed point set of a torus action on a space lifts to the full cohomology algebra of the space.

Keywords:Springer variety, Weyl group action, equivariant cohomology
Categories:14M15, 14F43, 55N91

18. CMB 2017 (vol 60 pp. 747)

Huang, Yanhe; Sottile, Frank; Zelenko, Igor
Injectivity of Generalized Wronski Maps
We study linear projections on Plücker space whose restriction to the Grassmannian is a non-trivial branched cover. When an automorphism of the Grassmannian preserves the fibers, we show that the Grassmannian is necessarily of $m$-dimensional linear subspaces in a symplectic vector space of dimension $2m$, and the linear map is the Lagrangian involution. The Wronski map for a self-adjoint linear differential operator and pole placement map for symmetric linear systems are natural examples.

Keywords:Wronski map, Plücker embedding, curves in Lagrangian Grassmannian, self-adjoint linear differential operator, symmetric linear control system, pole placement map
Categories:14M15, 34A30, 93B55

19. CMB 2017 (vol 60 pp. 309)

Hein, Nickolas; Sottile, Frank; Zelenko, Igor
A Congruence Modulo Four for Real Schubert Calculus with Isotropic Flags
We previously obtained a congruence modulo four for the number of real solutions to many Schubert problems on a square Grassmannian given by osculating flags. Here, we consider Schubert problems given by more general isotropic flags, and prove this congruence modulo four for the largest class of Schubert problems that could be expected to exhibit this congruence.

Keywords:Lagrangian Grassmannian, Wronski map, Shapiro Conjecture
Categories:14N15, 14P99

20. CMB 2017 (vol 60 pp. 225)

Bahmanpour, Kamal; Naghipour, Reza
Faltings' Finiteness Dimension of Local Cohomology Modules Over Local Cohen-Macaulay Rings
Let $(R, \frak m)$ denote a local Cohen-Macaulay ring and $I$ a non-nilpotent ideal of $R$. The purpose of this article is to investigate Faltings' finiteness dimension $f_I(R)$ and equidimensionalness of certain homomorphic image of $R$. As a consequence we deduce that $f_I(R)=\operatorname{max}\{1, \operatorname{ht} I\}$ and if $\operatorname{mAss}_R(R/I)$ is contained in $\operatorname{Ass}_R(R)$, then the ring $R/ I+\cup_{n\geq 1}(0:_RI^n)$ is equidimensional of dimension $\dim R-1$. Moreover, we will obtain a lower bound for injective dimension of the local cohomology module $H^{\operatorname{ht} I}_I(R)$, in the case $(R, \frak m)$ is a complete equidimensional local ring.

Keywords:Cohen Macaulay ring, equidimensional ring, finiteness dimension, local cohomology
Categories:13D45, 14B15

21. CMB 2017 (vol 61 pp. 166)

Miranda-Neto, Cleto B.
A module-theoretic characterization of algebraic hypersurfaces
In this note we prove the following surprising characterization: if $X\subset {\mathbb A}^n$ is an (embedded, non-empty, proper) algebraic variety defined over a field $k$ of characteristic zero, then $X$ is a hypersurface if and only if the module $T_{{\mathcal O}_{{\mathbb A}^n}/k}(X)$ of logarithmic vector fields of $X$ is a reflexive ${\mathcal O}_{{\mathbb A}^n}$-module. As a consequence of this result, we derive that if $T_{{\mathcal O}_{{\mathbb A}^n}/k}(X)$ is a free ${\mathcal O}_{{\mathbb A}^n}$-module, which is shown to be equivalent to the freeness of the $t$th exterior power of $T_{{\mathcal O}_{{\mathbb A}^n}/k}(X)$ for some (in fact, any) $t\leq n$, then necessarily $X$ is a Saito free divisor.

Keywords:hypersurface, logarithmic vector field, logarithmic derivation, free divisor
Categories:14J70, 13N15, 32S22, 13C05, 13C10, 14N20, , , , , 14C20, 32M25

22. CMB 2017 (vol 60 pp. 329)

Le Fourn, Samuel
Nonvanishing of Central Values of $L$-functions of Newforms in $S_2 (\Gamma_0 (dp^2))$ Twisted by Quadratic Characters
We prove that for $d \in \{ 2,3,5,7,13 \}$ and $K$ a quadratic (or rational) field of discriminant $D$ and Dirichlet character $\chi$, if a prime $p$ is large enough compared to $D$, there is a newform $f \in S_2(\Gamma_0(dp^2))$ with sign $(+1)$ with respect to the Atkin-Lehner involution $w_{p^2}$ such that $L(f \otimes \chi,1) \neq 0$. This result is obtained through an estimate of a weighted sum of twists of $L$-functions which generalises a result of Ellenberg. It relies on the approximate functional equation for the $L$-functions $L(f \otimes \chi, \cdot)$ and a Petersson trace formula restricted to Atkin-Lehner eigenspaces. An application of this nonvanishing theorem will be given in terms of existence of rank zero quotients of some twisted jacobians, which generalises a result of Darmon and Merel.

Keywords:nonvanishing of $L$-functions of modular forms, Petersson trace formula, rank zero quotients of jacobians
Categories:14J15, 11F67

23. CMB 2016 (vol 60 pp. 510)

Haase, Christian; Hofmann, Jan
Convex-normal (Pairs of) Polytopes
In 2012 Gubeladze (Adv. Math. 2012) introduced the notion of $k$-convex-normal 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 $(k+1)$-convex-normality (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

24. CMB 2016 (vol 60 pp. 613)

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 $(r-1)n^2 + 1$.

Keywords:determinantal hypersurface, matrix invariant, $q$-binomial coefficient
Categories:14M12, 15A22, 05A10

25. 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 S-equivalence classes of semistable bundles of rank 2 with trivial determinant.

Keywords:Hecke curve, (0,1) stable bundle
Category:14D21
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