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1. CJM Online first

Xia, Eugene Z.
The algebraic de Rham cohomology of representation varieties
The $\operatorname{SL}(2,\mathbb C)$-representation varieties of punctured surfaces form natural families parameterized by monodromies at the punctures. In this paper, we compute the loci where these varieties are singular for the cases of one-holed and two-holed tori and the four-holed sphere. We then compute the de Rham cohomologies of these varieties of the one-holed torus and the four-holed sphere when the varieties are smooth via the Grothendieck theorem. Furthermore, we produce the explicit Gauss-Manin connection on the natural family of the smooth $\operatorname{SL}(2,\mathbb C)$-representation varieties of the one-holed torus.

Keywords:surface, algebraic group, representation variety, de Rham cohomology
Categories:14H10, 13D03, 14F40, 14H24, 14Q10, 14R20

2. CJM Online first

Favacchio, Giuseppe; Guardo, Elena
The minimal free resolution of fat almost complete intersections in $\mathbb{P}^1\times \mathbb{P}^1$
A current research theme is to compare symbolic powers of an ideal $I$ with the regular powers of $I$. In this paper, we focus on the case that $I=I_X$ is an ideal defining an almost complete intersection (ACI) set of points $X$ in $\mathbb{P}^1 \times \mathbb{P}^1$. In particular, we describe a minimal free bigraded resolution of a non arithmetically Cohen-Macaulay (also non homogeneous) set $\mathcal Z$ of fat points whose support is an ACI, generalizing a result of S. Cooper et al. for homogeneous sets of triple points. We call $\mathcal Z$ a fat ACI. We also show that its symbolic and ordinary powers are equal, i.e, $I_{\mathcal Z}^{(m)}=I_{\mathcal Z}^{m}$ for any $m\geq 1.$

Keywords:points in $\mathbb{P}^1\times \mathbb{P}^1$, symbolic powers, resolution, arithmetically Cohen-Macaulay
Categories:13C40, 13F20, 13A15, 14C20, 14M05

3. CJM Online first

Choi, Suyoung; Park, Hanchul
Wedge operations and torus symmetries II
A fundamental idea in toric topology is that classes of manifolds with well-behaved torus actions (simply, toric spaces) are classified by pairs of simplicial complexes and (non-singular) characteristic maps. The authors in their previous paper provided a new way to find all characteristic maps on a simplicial complex $K(J)$ obtainable by a sequence of wedgings from $K$. The main idea was that characteristic maps on $K$ theoretically determine all possible characteristic maps on a wedge of $K$. In this work, we further develop our previous work for classification of toric spaces. For a star-shaped simplicial sphere $K$ of dimension $n-1$ with $m$ vertices, the Picard number $\operatorname{Pic}(K)$ of $K$ is $m-n$. We refer to $K$ as a seed if $K$ cannot be obtained by wedgings. First, we show that, for a fixed positive integer $\ell$, there are at most finitely many seeds of Picard number $\ell$ supporting characteristic maps. As a corollary, the conjecture proposed by V.V. Batyrev in 1991 is solved affirmatively. Second, we investigate a systematic method to find all characteristic maps on $K(J)$ using combinatorial objects called (realizable) puzzles that only depend on a seed $K$. These two facts lead to a practical way to classify the toric spaces of fixed Picard number.

Keywords:puzzle, toric variety, simplicial wedge, characteristic map
Categories:57S25, 14M25, 52B11, 13F55, 18A10

4. CJM 2016 (vol 69 pp. 241)

Adamus, Janusz; Seyedinejad, Hadi
Finite Determinacy and Stability of Flatness of Analytic Mappings
It is proved that flatness of an analytic mapping germ from a complete intersection is determined by its sufficiently high jet. As a consequence, one obtains finite determinacy of complete intersections. It is also shown that flatness and openness are stable under deformations.

Keywords:finite determinacy, stability, flatness, openness, complete intersection
Categories:58K40, 58K25, 32S05, 58K20, 32S30, 32B99, 32C05, 13B40

5. CJM 2014 (vol 67 pp. 923)

Pan, Ivan Edgardo; Simis, Aron
Cremona Maps of de Jonquières Type
This paper is concerned with suitable generalizations of a plane de Jonquières map to higher dimensional space $\mathbb{P}^n$ with $n\geq 3$. For each given point of $\mathbb{P}^n$ there is a subgroup of the entire Cremona group of dimension $n$ consisting of such maps. One studies both geometric and group-theoretical properties of this notion. In the case where $n=3$ one describes an explicit set of generators of the group and gives a homological characterization of a basic subgroup thereof.

Keywords:Cremona map, de Jonquières map, Cremona group, minimal free resolution
Categories:14E05, 13D02, 13H10, 14E07, 14M05, 14M25

6. CJM 2014 (vol 67 pp. 1024)

Ashraf, Samia; Azam, Haniya; Berceanu, Barbu
Representation Stability of Power Sets and Square Free Polynomials
The symmetric group $\mathcal{S}_n$ acts on the power set $\mathcal{P}(n)$ and also on the set of square free polynomials in $n$ variables. These two related representations are analyzed from the stability point of view. An application is given for the action of the symmetric group on the cohomology of the pure braid group.

Keywords:symmetric group modules, square free polynomials, representation stability, Arnold algebra
Categories:20C30, 13A50, 20F36, 55R80

7. 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

8. 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

9. 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

10. 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

11. CJM 2010 (vol 62 pp. 1131)

Kleppe, Jan O.
Moduli Spaces of Reflexive Sheaves of Rank 2
Let $\mathcal{F}$ be a coherent rank $2$ sheaf on a scheme $Y \subset \mathbb{P}^{n}$ of dimension at least two and let $X \subset Y$ be the zero set of a section $\sigma \in H^0(\mathcal{F})$. In this paper, we study the relationship between the functor that deforms the pair $(\mathcal{F},\sigma)$ and the two functors that deform $\mathcal{F}$ on $Y$, and $X$ in $Y$, respectively. By imposing some conditions on two forgetful maps between the functors, we prove that the scheme structure of \emph{e.g.,} the moduli scheme ${\rm M_Y}(P)$ of stable sheaves on a threefold $Y$ at $(\mathcal{F})$, and the scheme structure at $(X)$ of the Hilbert scheme of curves on $Y$ become closely related. Using this relationship, we get criteria for the dimension and smoothness of $ {\rm M_{Y}}(P)$ at $(\mathcal{F})$, without assuming $ {\textrm{Ext}^2}(\mathcal{F} ,\mathcal{F} ) = 0$. For reflexive sheaves on $Y=\mathbb{P}^{3}$ whose deficiency module $M = H_{*}^1(\mathcal{F})$ satisfies $ {_{0}\! \textrm{Ext}^2}(M ,M ) = 0 $ (\emph{e.g.,} of diameter at most 2), we get necessary and sufficient conditions of unobstructedness that coincide in the diameter one case. The conditions are further equivalent to the vanishing of certain graded Betti numbers of the free graded minimal resolution of $H_{*}^0(\mathcal{F})$. Moreover, we show that every irreducible component of ${\rm M}_{\mathbb{P}^{3}}(P)$ containing a reflexive sheaf of diameter one is reduced (generically smooth) and we compute its dimension. We also determine a good lower bound for the dimension of any component of ${\rm M}_{\mathbb{P}^{3}}(P)$ that contains a reflexive stable sheaf with ``small'' deficiency module $M$.

Keywords:moduli space, reflexive sheaf, Hilbert scheme, space curve, Buchsbaum sheaf, unobstructedness, cup product, graded Betti numbers.xdvi
Categories:14C05, qqqqq14D22, 14F05, 14J10, 14H50, 14B10, 13D02, 13D07

12. CJM 2009 (vol 62 pp. 721)

Boocher, Adam; Daub, Michael; Johnson, Ryan K.; Lindo, H.; Loepp, S.; Woodard, Paul A.
Formal Fibers of Unique Factorization Domains
Let $(T,M)$ be a complete local (Noetherian) ring such that $\dim T\geq 2$ and $|T|=|T/M|$ and let $\{p_i\} _{i \in \mathcal I}$ be a collection of elements of T indexed by a set $\mathcal I$ so that $|\mathcal I | < |T|$. For each $i \in \mathcal{I}$, let $C_i$:={$Q_{i1}$,$\dots$,$Q_{in_i}$} be a set of nonmaximal prime ideals containing $p_i$ such that the $Q_{ij}$ are incomparable and $p_i\in Q_{jk}$ if and only if $i=j$. We provide necessary and sufficient conditions so that T is the ${\bf m}$-adic completion of a local unique factorization domain $(A, {\bf m})$, and for each $i \in \mathcal I$, there exists a unit $t_i$ of T so that $p_{i}t_i \in A$ and $C_i$ is the set of prime ideals $Q$ of $T$ that are maximal with respect to the condition that $Q \cap A = p_{i}t_{i}A$. We then use this result to construct a (nonexcellent) unique factorization domain containing many ideals for which tight closure and completion do not commute. As another application, we construct a unique factorization domain A most of whose formal fibers are geometrically regular.

Categories:13J10, 13J05

13. CJM 2009 (vol 61 pp. 888)

Novik, Isabella; Swartz, Ed
Face Ring Multiplicity via CM-Connectivity Sequences
The multiplicity conjecture of Herzog, Huneke, and Srinivasan is verified for the face rings of the following classes of simplicial complexes: matroid complexes, complexes of dimension one and two, and Gorenstein complexes of dimension at most four. The lower bound part of this conjecture is also established for the face rings of all doubly Cohen--Macaulay complexes whose 1-skeleton's connectivity does not exceed the codimension plus one as well as for all $(d-1)$-dimensional $d$-Cohen--Macaulay complexes. The main ingredient of the proofs is a new interpretation of the minimal shifts in the resolution of the face ring $\field[\Delta]$ via the Cohen--Macaulay connectivity of the skeletons of $\Delta$.

Categories:13F55, 52B05;, 13H15;, 13D02;, 05B35

14. CJM 2009 (vol 61 pp. 950)

Tange, Rudolf
Infinitesimal Invariants in a Function Algebra
Let $G$ be a reductive connected linear algebraic group over an algebraically closed field of positive characteristic and let $\g$ be its Lie algebra. First we extend a well-known result about the Picard group of a semi-simple group to reductive groups. Then we prove that if the derived group is simply connected and $\g$ satisfies a mild condition, the algebra $K[G]^\g$ of regular functions on $G$ that are invariant under the action of $\g$ derived from the conjugation action is a unique factorisation domain.

Categories:20G15, 13F15

15. CJM 2009 (vol 61 pp. 930)

Sidman, Jessica; Sullivant, Seth
Prolongations and Computational Algebra
We explore the geometric notion of prolongations in the setting of computational algebra, extending results of Landsberg and Manivel which relate prolongations to equations for secant varieties. We also develop methods for computing prolongations that are combinatorial in nature. As an application, we use prolongations to derive a new family of secant equations for the binary symmetric model in phylogenetics.

Categories:13P10, 14M99

16. CJM 2009 (vol 61 pp. 762)

D'Cruz, Clare; Puthenpurakal, Tony J.
The Hilbert Coefficients of the Fiber Cone and the $a$-Invariant of the Associated Graded Ring
Let $(A,\m)$ be a Noetherian local ring with infinite residue field and let $I$ be an ideal in $A$ and let $F(I) = \bigoplus_{n \geq 0}I^n/\m I^n$ be the fiber cone of $I$. We prove certain relations among the Hilbert coefficients $f_0(I),f_1(I), f_2(I)$ of $F(I)$ when the $a$-invariant of the associated graded ring $G(I)$ is negative.

Keywords:fiber cone, $a$-invariant, Hilbert coefficients of fiber cone
Categories:13A30, 13D40

17. CJM 2009 (vol 61 pp. 29)

Casanellas, M.
The Minimal Resolution Conjecture for Points on the Cubic Surface
In this paper we prove that a generalized version of the Minimal Resolution Conjecture given by Musta\c{t}\v{a} holds for certain general sets of points on a smooth cubic surface $X \subset \PP^3$. The main tool used is Gorenstein liaison theory and, more precisely, the relationship between the free resolutions of two linked schemes.

Categories:13D02, 13C40, 14M05, 14M07

18. CJM 2009 (vol 61 pp. 205)

Marshall, M.
Representations of Non-Negative Polynomials, Degree Bounds and Applications to Optimization
Natural sufficient conditions for a polynomial to have a local minimum at a point are considered. These conditions tend to hold with probability $1$. It is shown that polynomials satisfying these conditions at each minimum point have nice presentations in terms of sums of squares. Applications are given to optimization on a compact set and also to global optimization. In many cases, there are degree bounds for such presentations. These bounds are of theoretical interest, but they appear to be too large to be of much practical use at present. In the final section, other more concrete degree bounds are obtained which ensure at least that the feasible set of solutions is not empty.

Categories:13J30, 12Y05, 13P99, 14P10, 90C22

19. CJM 2009 (vol 61 pp. 76)

Christensen, Lars Winther; Holm, Henrik
Ascent Properties of Auslander Categories
Let $R$ be a homomorphic image of a Gorenstein local ring. Recent work has shown that there is a bridge between Auslander categories and modules of finite Gorenstein homological dimensions over $R$. We use Gorenstein dimensions to prove new results about Auslander categories and vice versa. For example, we establish base change relations between the Auslander categories of the source and target rings of a homomorphism $\varphi \colon R \to S$ of finite flat dimension.

Keywords:Auslander categories, Gorenstein dimensions, ascent properties, Auslander--Buchsbaum formulas
Categories:13D05, 13D07, 13D25

20. CJM 2008 (vol 60 pp. 721)

Adamus, J.; Bierstone, E.; Milman, P. D.
Uniform Linear Bound in Chevalley's Lemma
We obtain a uniform linear bound for the Chevalley function at a point in the source of an analytic mapping that is regular in the sense of Gabrielov. There is a version of Chevalley's lemma also along a fibre, or at a point of the image of a proper analytic mapping. We get a uniform linear bound for the Chevalley function of a closed Nash (or formally Nash) subanalytic set.

Keywords:Chevalley function, regular mapping, Nash subanalytic set
Categories:13J07, 32B20, 13J10, 32S10

21. CJM 2008 (vol 60 pp. 556)

Draisma, Jan; Kemper, Gregor; Wehlau, David
Polarization of Separating Invariants
We prove a characteristic free version of Weyl's theorem on polarization. Our result is an exact analogue of Weyl's theorem, the difference being that our statement is about separating invariants rather than generating invariants. For the special case of finite group actions we introduce the concept of \emph{cheap polarization}, and show that it is enough to take cheap polarizations of invariants of just one copy of a representation to obtain separating vector invariants for any number of copies. This leads to upper bounds on the number and degrees of separating vector invariants of finite groups.

Keywords:Jan Draisma, Gregor Kemper, David Wehlau
Categories:13A50, 14L24

22. CJM 2008 (vol 60 pp. 391)

Migliore, Juan C.
The Geometry of the Weak Lefschetz Property and Level Sets of Points
In a recent paper, F. Zanello showed that level Artinian algebras in 3 variables can fail to have the Weak Lefschetz Property (WLP), and can even fail to have unimodal Hilbert function. We show that the same is true for the Artinian reduction of reduced, level sets of points in projective 3-space. Our main goal is to begin an understanding of how the geometry of a set of points can prevent its Artinian reduction from having WLP, which in itself is a very algebraic notion. More precisely, we produce level sets of points whose Artinian reductions have socle types 3 and 4 and arbitrary socle degree $\geq 12$ (in the worst case), but fail to have WLP. We also produce a level set of points whose Artinian reduction fails to have unimodal Hilbert function; our example is based on Zanello's example. Finally, we show that a level set of points can have Artinian reduction that has WLP but fails to have the Strong Lefschetz Property. While our constructions are all based on basic double G-linkage, the implementations use very different methods.

Keywords:Weak Lefschetz Property, Strong Lefschetz Property, basic double G-linkage, level, arithmetically Gorenstein, arithmetically Cohen--Macaulay, socle type, socle degree, Artinian reduction
Categories:13D40, 13D02, 14C20, 13C40, 13C13, 14M05

23. CJM 2007 (vol 59 pp. 880)

van, John E.
Radical Ideals in Valuation Domains
An ideal $I$ of a ring $R$ is called a radical ideal if $I={\mathcalR}(R)$ where ${\mathcal R}$ is a radical in the sense of Kurosh--Amitsur. The main theorem of this paper asserts that if $R$ is a valuation domain, then a proper ideal $I$ of $R$ is a radical ideal if and only if $I$ is a distinguished ideal of $R$ (the latter property means that if $J$ and $K$ are ideals of $R$ such that $J\subset I\subset K$ then we cannot have $I/J\cong K/I$ as rings) and that such an ideal is necessarily prime. Examples are exhibited which show that, unlike prime ideals, distinguished ideals are not characterizable in terms of a property of the underlying value group of the valuation domain.

Categories:16N80, 13A18

24. CJM 2007 (vol 59 pp. 109)

Jayanthan, A. V.; Puthenpurakal, Tony J.; Verma, J. K.
On Fiber Cones of $\m$-Primary Ideals
Two formulas for the multiplicity of the fiber cone $F(I)=\bigoplus_{n=0}^{\infty} I^n/\m I^n$ of an $\m$-primary ideal of a $d$-dimensional Cohen--Macaulay local ring $(R,\m)$ are derived in terms of the mixed multiplicity $e_{d-1}(\m | I)$, the multiplicity $e(I)$, and superficial elements. As a consequence, the Cohen--Macaulay property of $F(I)$ when $I$ has minimal mixed multiplicity or almost minimal mixed multiplicity is characterized in terms of the reduction number of $I$ and lengths of certain ideals. We also characterize the Cohen--Macaulay and Gorenstein properties of fiber cones of $\m$-primary ideals with a $d$-generated minimal reduction $J$ satisfying $\ell(I^2/JI)=1$ or $\ell(I\m/J\m)=1.$

Keywords:fiber cones, mixed multiplicities, joint reductions, Cohen--Macaulay fiber cones, Gorenstein fiber cones, ideals having minimal and almost minimal mixed multiplicities
Categories:13H10, 13H15, 13A30, 13C15, 13A02

25. CJM 2005 (vol 57 pp. 1178)

Cutkosky, Steven Dale; Hà, Huy Tài; Srinivasan, Hema; Theodorescu, Emanoil
Asymptotic Behavior of the Length of Local Cohomology
Let $k$ be a field of characteristic 0, $R=k[x_1, \ldots, x_d]$ be a polynomial ring, and $\mm$ its maximal homogeneous ideal. Let $I \subset R$ be a homogeneous ideal in $R$. Let $\lambda(M)$ denote the length of an $R$-module $M$. In this paper, we show that $$ \lim_{n \to \infty} \frac{\l\bigl(H^0_{\mathfrak{m}}(R/I^n)\bigr)}{n^d} =\lim_{n \to \infty} \frac{\l\bigl(\Ext^d_R\bigl(R/I^n,R(-d)\bigr)\bigr)}{n^d} $$ always exists. This limit has been shown to be ${e(I)}/{d!}$ for $m$-primary ideals $I$ in a local Cohen--Macaulay ring, where $e(I)$ denotes the multiplicity of $I$. But we find that this limit may not be rational in general. We give an example for which the limit is an irrational number thereby showing that the lengths of these extention modules may not have polynomial growth.

Keywords:powers of ideals, local cohomology, Hilbert function, linear growth
Categories:13D40, 14B15, 13D45
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