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26. CJM 2014 (vol 67 pp. 667)

Nishinou, Takeo
Toric Degenerations, Tropical Curve, and Gromov-Witten Invariants of Fano Manifolds
In this paper, we give a tropical method for computing Gromov-Witten type invariants of Fano manifolds of special type. This method applies to those Fano manifolds which admit toric degenerations to toric Fano varieties with singularities allowing small resolutions. Examples include (generalized) flag manifolds of type A, and some moduli space of rank two bundles on a genus two curve.

Keywords:Fano varieties, Gromov-Witten invariants, tropical curves
Category:14J45

27. CJM 2014 (vol 67 pp. 286)

Bell, Jason P.; Lagarias, Jeffrey C.
A Skolem-Mahler-Lech Theorem for Iterated Automorphisms of $K$-algebras
This paper proves a commutative algebraic extension of a generalized Skolem-Mahler-Lech theorem due to the first author. Let $A$ be a finitely generated commutative $K$-algebra over a field of characteristic $0$, and let $\sigma$ be a $K$-algebra automorphism of $A$. Given ideals $I$ and $J$ of $A$, we show that the set $S$ of integers $m$ such that $\sigma^m(I) \supseteq J$ is a finite union of complete doubly infinite arithmetic progressions in $m$, up to the addition of a finite set. Alternatively, this result states that for an affine scheme $X$ of finite type over $K$, an automorphism $\sigma \in \operatorname{Aut}_K(X)$, and $Y$ and $Z$ any two closed subschemes of $X$, the set of integers $m$ with $\sigma^m(Z ) \subseteq Y$ is as above. The paper presents examples showing that this result may fail to hold if the affine scheme $X$ is not of finite type, or if $X$ is of finite type but the field $K$ has positive characteristic.

Keywords:automorphisms, endomorphisms, affine space, commutative algebras, Skolem-Mahler-Lech theorem
Categories:11D45, 14R10, 11Y55, 11D88

28. CJM 2013 (vol 66 pp. 1250)

Feigin, Evgeny; Finkelberg, Michael; Littelmann, Peter
Symplectic Degenerate Flag Varieties
A simple finite dimensional module $V_\lambda$ of a simple complex algebraic group $G$ is naturally endowed with a filtration induced by the PBW-filtration of $U(\mathrm{Lie}\, G)$. The associated graded space $V_\lambda^a$ is a module for the group $G^a$, which can be roughly described as a semi-direct product of a Borel subgroup of $G$ and a large commutative unipotent group $\mathbb{G}_a^M$. In analogy to the flag variety $\mathcal{F}_\lambda=G.[v_\lambda]\subset \mathbb{P}(V_\lambda)$, we call the closure $\overline{G^a.[v_\lambda]}\subset \mathbb{P}(V_\lambda^a)$ of the $G^a$-orbit through the highest weight line the degenerate flag variety $\mathcal{F}^a_\lambda$. In general this is a singular variety, but we conjecture that it has many nice properties similar to that of Schubert varieties. In this paper we consider the case of $G$ being the symplectic group. The symplectic case is important for the conjecture because it is the first known case where even for fundamental weights $\omega$ the varieties $\mathcal{F}^a_\omega$ differ from $\mathcal{F}_\omega$. We give an explicit construction of the varieties $Sp\mathcal{F}^a_\lambda$ and construct desingularizations, similar to the Bott-Samelson resolutions in the classical case. We prove that $Sp\mathcal{F}^a_\lambda$ are normal locally complete intersections with terminal and rational singularities. We also show that these varieties are Frobenius split. Using the above mentioned results, we prove an analogue of the Borel-Weil theorem and obtain a $q$-character formula for the characters of irreducible $Sp_{2n}$-modules via the Atiyah-Bott-Lefschetz fixed points formula.

Keywords:Lie algebras, flag varieties, symplectic groups, representations
Categories:14M15, 22E46

29. CJM 2013 (vol 66 pp. 1305)

Koskivirta, Jean-Stefan
Congruence Relations for Shimura Varieties Associated with $GU(n-1,1)$
We prove the congruence relation for the mod-$p$ reduction of Shimura varieties associated to a unitary similitude group $GU(n-1,1)$ over $\mathbb{Q}$, when $p$ is inert and $n$ odd. The case when $n$ is even was obtained by T. Wedhorn and O. B?ltel, as a special case of a result of B. Moonen, when the $\mu$-ordinary locus of the $p$-isogeny space is dense. This condition fails in our case. We show that every supersingular irreducible component of the special fiber of $p\textrm{-}\mathscr{I}sog$ is annihilated by a degree one polynomial in the Frobenius element $F$, which implies the congruence relation.

Keywords:Shimura varieties, congruence relation
Categories:11G18, 14G35, 14K10

30. CJM 2013 (vol 66 pp. 505)

Arapura, Donu
Hodge Theory of Cyclic Covers Branched over a Union of Hyperplanes
Suppose that $Y$ is a cyclic cover of projective space branched over a hyperplane arrangement $D$, and that $U$ is the complement of the ramification locus in $Y$. The first theorem implies that the Beilinson-Hodge conjecture holds for $U$ if certain multiplicities of $D$ are coprime to the degree of the cover. For instance this applies when $D$ is reduced with normal crossings. The second theorem shows that when $D$ has normal crossings and the degree of the cover is a prime number, the generalized Hodge conjecture holds for any toroidal resolution of $Y$. The last section contains some partial extensions to more general nonabelian covers.

Keywords:Hodge cycles, hyperplane arrangements
Category:14C30

31. CJM 2013 (vol 66 pp. 1225)

Cortadellas Benítez, Teresa; D'Andrea, Carlos
Minimal Generators of the Defining Ideal of the Rees Algebra Associated with a Rational Plane Parametrization with $\mu=2$
We exhibit a set of minimal generators of the defining ideal of the Rees Algebra associated with the ideal of three bivariate homogeneous polynomials parametrizing a proper rational curve in projective plane, having a minimal syzygy of degree 2.

Keywords:Rees Algebras, rational plane curves, minimal generators
Categories:13A30, 14H50

32. CJM 2013 (vol 66 pp. 924)

Stankewicz, James
Twists of Shimura Curves
Consider a Shimura curve $X^D_0(N)$ over the rational numbers. We determine criteria for the twist by an Atkin-Lehner involution to have points over a local field. As a corollary we give a new proof of the theorem of Jordan-Livné on $\mathbf{Q}_p$ points when $p\mid D$ and for the first time give criteria for $\mathbf{Q}_p$ points when $p\mid N$. We also give congruence conditions for roots modulo $p$ of Hilbert class polynomials.

Keywords:Shimura curves, complex multiplication, modular curves, elliptic curves
Categories:11G18, 14G35, 11G15, 11G10

33. CJM 2013 (vol 65 pp. 1125)

Vandenbergen, Nicolas
On the Global Structure of Special Cycles on Unitary Shimura Varieties
In this paper, we study the reduced loci of special cycles on local models of the Shimura variety for $\operatorname{GU}(1,n-1)$. Those special cycles are defined by Kudla and Rapoport. We explicitly compute the irreducible components of the reduced locus of a single special cycle, as well as of an arbitrary intersection of special cycles, and their intersection behaviour in terms of Bruhat-Tits theory. Furthermore, as an application of our results, we prove the connectedness of arbitrary intersections of special cycles, as conjectured by Kudla and Rapoport.

Keywords:Shimura varieties, local models, special cycles
Category:14G35

34. CJM 2013 (vol 66 pp. 1167)

Rotger, Victor; de Vera-Piquero, Carlos
Galois Representations Over Fields of Moduli and Rational Points on Shimura Curves
The purpose of this note is introducing a method for proving the existence of no rational points on a coarse moduli space $X$ of abelian varieties over a given number field $K$, in cases where the moduli problem is not fine and points in $X(K)$ may not be represented by an abelian variety (with additional structure) admitting a model over the field $K$. This is typically the case when the abelian varieties that are being classified have even dimension. The main idea, inspired on the work of Ellenberg and Skinner on the modularity of $\mathbb{Q}$-curves, is that to a point $P=[A]\in X(K)$ represented by an abelian variety $A/\bar K$ one may still attach a Galois representation of $\operatorname{Gal}(\bar K/K)$ with values in the quotient group $\operatorname{GL}(T_\ell(A))/\operatorname{Aut}(A)$, provided $\operatorname{Aut}(A)$ lies in the centre of $\operatorname{GL}(T_\ell(A))$. We exemplify our method in the cases where $X$ is a Shimura curve over an imaginary quadratic field or an Atkin-Lehner quotient over $\mathbb{Q}$.

Keywords:Shimura curves, rational points, Galois representations, Hasse principle, Brauer-Manin obstruction
Categories:11G18, 14G35, 14G05

35. CJM 2012 (vol 66 pp. 3)

Abdesselam, Abdelmalek; Chipalkatti, Jaydeep
On Hilbert Covariants
Let $F$ denote a binary form of order $d$ over the complex numbers. If $r$ is a divisor of $d$, then the Hilbert covariant $\mathcal{H}_{r,d}(F)$ vanishes exactly when $F$ is the perfect power of an order $r$ form. In geometric terms, the coefficients of $\mathcal{H}$ give defining equations for the image variety $X$ of an embedding $\mathbf{P}^r \hookrightarrow \mathbf{P}^d$. In this paper we describe a new construction of the Hilbert covariant; and simultaneously situate it into a wider class of covariants called the Göttingen covariants, all of which vanish on $X$. We prove that the ideal generated by the coefficients of $\mathcal{H}$ defines $X$ as a scheme. Finally, we exhibit a generalisation of the Göttingen covariants to $n$-ary forms using the classical Clebsch transfer principle.

Keywords:binary forms, covariants, $SL_2$-representations
Categories:14L30, 13A50

36. CJM 2012 (vol 65 pp. 823)

Guardo, Elena; Harbourne, Brian; Van Tuyl, Adam
Symbolic Powers Versus Regular Powers of Ideals of General Points in $\mathbb{P}^1 \times \mathbb{P}^1$
Recent work of Ein-Lazarsfeld-Smith and Hochster-Huneke raised the problem of which symbolic powers of an ideal are contained in a given ordinary power of the ideal. Bocci-Harbourne developed methods to address this problem, which involve asymptotic numerical characters of symbolic powers of the ideals. Most of the work done up to now has been done for ideals defining 0-dimensional subschemes of projective space. Here we focus on certain subschemes given by a union of lines in $\mathbb{P}^3$ which can also be viewed as points in $\mathbb{P}^1 \times \mathbb{P}^1$. We also obtain results on the closely related problem, studied by Hochster and by Li-Swanson, of determining situations for which each symbolic power of an ideal is an ordinary power.

Keywords:symbolic powers, multigraded, points
Categories:13F20, 13A15, 14C20

37. CJM 2012 (vol 65 pp. 1020)

Goulden, I. P.; Guay-Paquet, Mathieu; Novak, Jonathan
Monotone Hurwitz Numbers in Genus Zero
Hurwitz numbers count branched covers of the Riemann sphere with specified ramification data, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of these branched covers related to the expansion of complete symmetric functions in the Jucys-Murphy elements, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra-Itzykson-Zuber integral. In this paper we begin a detailed study of monotone Hurwitz numbers. We prove two results that are reminiscent of those for classical Hurwitz numbers. The first is the monotone join-cut equation, a partial differential equation with initial conditions that characterizes the generating function for monotone Hurwitz numbers in arbitrary genus. The second is our main result, in which we give an explicit formula for monotone Hurwitz numbers in genus zero.

Keywords:Hurwitz numbers, matrix models, enumerative geometry
Categories:05A15, 14E20, 15B52

38. CJM 2012 (vol 65 pp. 905)

Thompson, Alan
Explicit Models for Threefolds Fibred by K3 Surfaces of Degree Two
We consider threefolds that admit a fibration by K3 surfaces over a nonsingular curve, equipped with a divisorial sheaf that defines a polarisation of degree two on the general fibre. Under certain assumptions on the threefold we show that its relative log canonical model exists and can be explicitly reconstructed from a small set of data determined by the original fibration. Finally we prove a converse to the above statement: under certain assumptions, any such set of data determines a threefold that arises as the relative log canonical model of a threefold admitting a fibration by K3 surfaces of degree two.

Keywords:threefold, fibration, K3 surface
Categories:14J30, 14D06, 14E30, 14J28

39. CJM 2012 (vol 65 pp. 634)

Mezzetti, Emilia; Miró-Roig, Rosa M.; Ottaviani, Giorgio
Laplace Equations and the Weak Lefschetz Property
We prove that $r$ independent homogeneous polynomials of the same degree $d$ become dependent when restricted to any hyperplane if and only if their inverse system parameterizes a variety whose $(d-1)$-osculating spaces have dimension smaller than expected. This gives an equivalence between an algebraic notion (called Weak Lefschetz Property) and a differential geometric notion, concerning varieties which satisfy certain Laplace equations. In the toric case, some relevant examples are classified and as byproduct we provide counterexamples to Ilardi's conjecture.

Keywords:osculating space, weak Lefschetz property, Laplace equations, toric threefold
Categories:13E10, 14M25, 14N05, 14N15, 53A20

40. CJM 2012 (vol 65 pp. 575)

Kallel, Sadok; Taamallah, Walid
The Geometry and Fundamental Group of Permutation Products and Fat Diagonals
Permutation products and their various ``fat diagonal'' subspaces are studied from the topological and geometric point of view. We describe in detail the stabilizer and orbit stratifications related to the permutation action, producing a sharp upper bound for its depth and then paying particular attention to the geometry of the diagonal stratum. We write down an expression for the fundamental group of any permutation product of a connected space $X$ having the homotopy type of a CW complex in terms of $\pi_1(X)$ and $H_1(X;\mathbb{Z})$. We then prove that the fundamental group of the configuration space of $n$-points on $X$, of which multiplicities do not exceed $n/2$, coincides with $H_1(X;\mathbb{Z})$. Further results consist in giving conditions for when fat diagonal subspaces of manifolds can be manifolds again. Various examples and homological calculations are included.

Keywords:symmetric products, fundamental group, orbit stratification
Categories:14F35, 57F80

41. CJM 2012 (vol 65 pp. 961)

Aholt, Chris; Sturmfels, Bernd; Thomas, Rekha
A Hilbert Scheme in Computer Vision
Multiview geometry is the study of two-dimensional images of three-dimensional scenes, a foundational subject in computer vision. We determine a universal Gröbner basis for the multiview ideal of $n$ generic cameras. As the cameras move, the multiview varieties vary in a family of dimension $11n-15$. This family is the distinguished component of a multigraded Hilbert scheme with a unique Borel-fixed point. We present a combinatorial study of ideals lying on that Hilbert scheme.

Keywords:multigraded Hilbert Scheme, computer vision, monomial ideal, Groebner basis, generic initial ideal
Categories:14N, 14Q, 68

42. CJM 2012 (vol 64 pp. 1222)

Bobiński, Grzegorz
Normality of Maximal Orbit Closures for Euclidean Quivers
Let $\Delta$ be an Euclidean quiver. We prove that the closures of the maximal orbits in the varieties of representations of $\Delta$ are normal and Cohen--Macaulay (even complete intersections). Moreover, we give a generalization of this result for the tame concealed-canonical algebras.

Keywords:normal variety, complete intersection, Euclidean quiver, concealed-canonical algebra
Categories:16G20, 14L30

43. CJM 2012 (vol 65 pp. 721)

Adamus, Janusz; Randriambololona, Serge; Shafikov, Rasul
Tameness of Complex Dimension in a Real Analytic Set
Given a real analytic set $X$ in a complex manifold and a positive integer $d$, denote by $\mathcal A^d$ the set of points $p$ in $X$ at which there exists a germ of a complex analytic set of dimension $d$ contained in $X$. It is proved that $\mathcal A^d$ is a closed semianalytic subset of $X$.

Keywords:complex dimension, finite type, semianalytic set, tameness
Categories:32B10, 32B20, 32C07, 32C25, 32V15, 32V40, 14P15

44. CJM 2012 (vol 65 pp. 544)

Deitmar, Anton; Horozov, Ivan
Iterated Integrals and Higher Order Invariants
We show that higher order invariants of smooth functions can be written as linear combinations of full invariants times iterated integrals. The non-uniqueness of such a presentation is captured in the kernel of the ensuing map from the tensor product. This kernel is computed explicitly. As a consequence, it turns out that higher order invariants are a free module of the algebra of full invariants.

Keywords:higher order forms, iterated integrals
Categories:14F35, 11F12, 55D35, 58A10

45. CJM 2012 (vol 65 pp. 195)

Penegini, Matteo; Polizzi, Francesco
Surfaces with $p_g=q=2$, $K^2=6$, and Albanese Map of Degree $2$
We classify minimal surfaces of general type with $p_g=q=2$ and $K^2=6$ whose Albanese map is a generically finite double cover. We show that the corresponding moduli space is the disjoint union of three generically smooth irreducible components $\mathcal{M}_{Ia}$, $\mathcal{M}_{Ib}$, $\mathcal{M}_{II}$ of dimension $4$, $4$, $3$, respectively.

Keywords:surface of general type, abelian surface, Albanese map
Categories:14J29, 14J10

46. CJM 2012 (vol 65 pp. 120)

Francois, Georges; Hampe, Simon
Universal Families of Rational Tropical Curves
We introduce the notion of families of $n$-marked smooth rational tropical curves over smooth tropical varieties and establish a one-to-one correspondence between (equivalence classes of) these families and morphisms from smooth tropical varieties into the moduli space of $n$-marked abstract rational tropical curves $\mathcal{M}_{n}$.

Keywords:tropical geometry, universal family, rational curves, moduli space
Categories:14T05, 14D22

47. CJM 2011 (vol 64 pp. 1090)

Rosso, Daniele
Classic and Mirabolic Robinson-Schensted-Knuth Correspondence for Partial Flags
In this paper we first generalize to the case of partial flags a result proved both by Spaltenstein and by Steinberg that relates the relative position of two complete flags and the irreducible components of the flag variety in which they lie, using the Robinson-Schensted-Knuth correspondence. Then we use this result to generalize the mirabolic Robinson-Schensted-Knuth correspondence defined by Travkin, to the case of two partial flags and a line.

Keywords:partial flag varieties, RSK correspondence
Categories:14M15, 05A05

48. CJM 2011 (vol 64 pp. 1248)

Gärtner, Jérôme
Darmon's Points and Quaternionic Shimura Varieties
In this paper, we generalize a conjecture due to Darmon and Logan in an adelic setting. We study the relation between our construction and Kudla's works on cycles on orthogonal Shimura varieties. This relation allows us to conjecture a Gross-Kohnen-Zagier theorem for Darmon's points.

Keywords:elliptic curves, Stark-Heegner points, quaternionic Shimura varieties
Categories:11G05, 14G35, 11F67, 11G40

49. CJM 2011 (vol 64 pp. 3)

Boissière, Samuel
Automorphismes naturels de l'espace de Douady de points sur une surface
On établit quelques résultats généraux relatifs à la taille du groupe d'automorphismes de l'espace de Douady de points sur une surface, puis on étudie quelques propriétés des automorphismes provenant d'un automorphisme de la surface, en particulier leur action sur la cohomologie et la classification de leurs points fixes.

Keywords:Schéma de Hilbert, automorphismes, points fixes
Category:14C05

50. CJM 2011 (vol 64 pp. 1122)

Seveso, Marco Adamo
$p$-adic $L$-functions and the Rationality of Darmon Cycles
Darmon cycles are a higher weight analogue of Stark--Heegner points. They yield local cohomology classes in the Deligne representation associated with a cuspidal form on $\Gamma _{0}( N) $ of even weight $k_{0}\geq 2$. They are conjectured to be the restriction of global cohomology classes in the Bloch--Kato Selmer group defined over narrow ring class fields attached to a real quadratic field. We show that suitable linear combinations of them obtained by genus characters satisfy these conjectures. We also prove $p$-adic Gross--Zagier type formulas, relating the derivatives of $p$-adic $L$-functions of the weight variable attached to imaginary (resp. real) quadratic fields to Heegner cycles (resp. Darmon cycles). Finally we express the second derivative of the Mazur--Kitagawa $p$-adic $L$-function of the weight variable in terms of a global cycle defined over a quadratic extension of $\mathbb{Q}$.

Categories:11F67, 14G05
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