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26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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

36. 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

37. 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

38. 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

39. 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

40. 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

41. 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

42. 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 $(n-1)\cdot \operatorname{degree}(T(G/P))/n$. 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

43. 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

44. CMB 2009 (vol 52 pp. 493)

Artebani, Michela
A One-Dimensional 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 one-dimensional family of $K3$ surfaces associated to the locus of plane quartics with five nodes.

Keywords:genus three curves, $K3$ surfaces
Categories:14J28, 14J50, 14J10

45. 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

46. CMB 2009 (vol 52 pp. 224)

Ghiloni, Riccardo
Equations and Complexity for the Dubois--Efroymson 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

47. 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

48. CMB 2009 (vol 52 pp. 161)

Arcara, D.; Lee, Y.-P.
A New Tautological Relation in $\overline{\mathcal{M}}_{3,1}$ via the Invariance Constraint
A new tautological relation of $\overline{\mathcal{M}}_{3,1}$ in codimension 3 is derived and proved, using an invariance constraint from our previous work.

Category:14H10

49. 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

50. 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 Gelfand--Naimark representation theorem for commutative $C^\ast$-algebras. A noncommutative version of Gelfand--Naimark 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
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