Invariant hypersurfaces for derivations in positive characteristic

Let A be an integral k-algebra of finite type over an algebraically closed field k of characteristic p>0. Given a collection D of k-derivations on A, that we interpret as algebraic vector fields on , we study the group spanned by the hypersurfaces V(f) of X invariant under D modulo the rational first integrals of D. We prove that this group is always a finite dimensional F_p-vector space, and we give an estimate for its dimension. This is to be related to the results of Jouanolou and others on the number of hypersurfaces invariant under a foliation of codimension 1. As a application, given a k-algebra B between A^p and A, we show that the kernel of the pull-back morphism is a finite F_p-vector space. In particular, if A is a UFD, then the Picard group of B is finite.     

Nice derivations over principal ideal domains

In this paper we investigate to what extent the results of Z. Wang and D. Daigle on “nice derivations” of the polynomial ring $k[X,Y,Z]$ over a field k of characteristic zero extend to the polynomial ring $R[X,Y,Z]$ over a PID R, containing the field of rational numbers. One of our results shows that the kernel of a nice derivation on $k[X_1,X_2,X_3,X_4]$ of rank at most three is a polynomial ring over k.     

On the classification of non-normal cubic hypersurfaces

In this article we study the classification of non-normal cubic hypersurfaces over an algebraically closed field K of arbitrary characteristic. Let be an irreducible non-normal cubic hypersurface. If r≥5, then X is necessarily a cone (Remark 2.3). In view of this fact it suffices to classify irreducible non-normal cubic hypersurfaces for r≤4. We prove that there are precisely five non-normal cubic equations (resp. six non-normal cubic equations) when (resp. when is either 2 or 3), up to projective equivalence. Also we describe the normalization of X in detail.     

Smoothness of the images of members of a linear system under an endomorphism of the affine plane

Let φ=(f,g) be an endomorphism of the affine plane C² defined by two polynomials f,g∈C[x,y] and let Λ={C_b∣b∈C} be the pencil of lines C_b defined by x=b. We shall consider the smoothness criterion of the image curve φ(C_b). The hypersurface V whose coordinate ring is C[x,f,g] and the normalization of V will play interesting roles in analyzing the properties of the set φ(Λ)={φ(C_b)∣b∈C}.     

On the recognition problem of quasi-homogeneous locally nilpotent derivations in dimension three

We give a criterion to decide if a given w-homogeneous derivation on A?k[X₁,X₂,X₃] is locally nilpotent. We deduce an algorithm which decides if a k-subalgebra of A, which is finitely generated by w-homogeneous elements, is the kernel of some locally nilpotent derivation.     

A note on residual variables of an affine fibration

In a recent paper [7], M.E. Kahoui has shown that if R   is a polynomial ring over C$\mathbb{C}$, A   an A³${\mathbb{A}}^3$-fibration over R, and W a residual variable of A then A   is stably polynomial over R[W]$R[W]$. In this article we show that the above result holds over any Noetherian domain R   provided the module of differentials _ΩR(A)${\Omega }_R(A)$ of the affine fibration A (which is necessarily a projective A-module by a theorem of Asanuma) is a stably free A-module.     

The limiting behavior on the restriction of divisor classes to hypersurfaces

Let A be an excellent local normal domain and {f_n}_n=1^∞ a sequence of elements lying in successively higher powers of the maximal ideal, such that each hypersurface A/f_nA satisfies R₁. We investigate the injectivity of the maps Cl(A)→Cl((A/f_nA)′), where (A/f_nA)′ represents the integral closure. The first result shows that no non-trivial divisor class can lie in every kernel. Secondly, when A is, in addition, an isolated singularity containing a field of characteristic zero, dim A?4, and A has a small Cohen-Macaulay module, then we show that there is an integer N>0 such that if , then Cl(A)→Cl((A/f_nA)′) is injective. We substantiate these results with a general construction that provides a large collection of examples.     

Connections on modules over singularities of finite CM representation type

Let A be a commutative k-algebra, where k is an algebraically closed field of characteristic 0, and let M be an A-module. We consider the following question: Under what conditions is it possible to find a connection on M?We consider the maximal Cohen-Macaulay (MCM) modules over complete CM algebras that are isolated singularities, and usually assume that the singularities have finite CM representation type. It is known that any MCM module over a simple singularity of dimension d≤2 admits an integrable connection. We prove that an MCM module over a simple singularity of dimension d≥3 admits a connection if and only if it is free. Among singularities of finite CM representation type, we find examples of curves with MCM modules that do not admit connections, and threefolds with non-free MCM modules that admit connections.Let A be a singularity not necessarily of finite CM representation type, and consider the condition that A is a Gorenstein curve or a -Gorenstein singularity of dimension d≥2. We show that this condition is sufficient for the canonical module ω_A to admit an integrable connection, and conjecture that it is also necessary. In support of the conjecture, we show that if A is a monomial curve singularity, then the canonical module ω_A admits an integrable connection if and only if A is Gorenstein.     

Singularities of the secant variety

We give positivity conditions on the embedding of a smooth variety which guarantee the normality of the secant variety, generalizing earlier results of the author and others. We also give classes of secant varieties satisfying the Hodge conjecture as well as a result on the singular locus of degenerate secant varieties.     

Images of locally finite derivations of polynomial algebras in two variables

In this paper we show that the image of any locally finite k-derivation of the polynomial algebra k[x,y] in two variables over a field k of characteristic zero is a Mathieu subspace. We also show that the two-dimensional Jacobian conjecture is equivalent to the statement that the image of every k-derivation D of k[x,y] such that and is a Mathieu subspace of k[x,y].     

On the curvature of algebraic hypersurfaces

Using algebraic residue theory, we try to generalize a theorem of Chasles about osculating circles of plane algebraic curves to algebraic hypersurfaces over algebraically closed fields of characteristic zero.     

Linear subspaces,symbolic powers and Nagata type conjectures

Inspired by results of Guardo, Van Tuyl and the second author for lines in P³${\mathbb{P}}^3$, we develop asymptotic upper bounds for the least degree of a homogeneous form vanishing to order at least m on a union of disjoint r  -dimensional planes in Pⁿ${\mathbb{P}}^n$ for n?2r+1$n?2r+1$. These considerations lead to new conjectures that suggest that the well known conjecture of Nagata for points in P²${\mathbb{P}}^2$ is not an exotic statement but rather a manifestation of a much more general phenomenon which seems to have been overlooked so far.     

Generic rings for Picard-Vessiot extensions and generic differential equations

Let G be an observable subgroup of GL_n. We produce an extension of differential commutative rings generic for Picard-Vessiot extensions with group G.     

Discriminant loci of ample and spanned line bundles

Let (X,L,V) be a triplet where X is an irreducible smooth complex projective variety, L is an ample and spanned line bundle on X and V⊆H⁰(X,L) spans L. The discriminant locus D(X,V)⊂|V| is the algebraic subset of singular elements of |V|. We study the components of D(X,V) in connection with the jumping sets of (X,V), generalizing the classical biduality theorem. We also deal with the degree of the discriminant (codegree of (X,L,V)) giving some bounds on it and classifying curves and surfaces of codegree 2 and 3. We exclude the possibility for the codegree to be 1. Significant examples are provided.     

Cancellation for two-dimensional unique factorization domains

We show that if A and B are finitely generated two-dimensional unique factorization domains over an algebraically closed field, then A[x]≅B[x] implies A≅B. The proof is an application of an algebraic technique involving the AK invariant which has previously been used to obtain other cancellation theorems.     

Derivations of polynomial algebras without Darboux polynomials

We present several new examples of homogeneous derivations of a polynomial ring k[X]=k[x₁,…,x_n] over a field k of characteristic zero without Darboux polynomials. Using a modification of a result of Shamsuddin, we produce these examples by induction on the number n of variables, thus more easily than the previously known example multidimensional Jouanolou systems of ?o?a?dek.     

Generic residual intersections and intersection numbers of movable components

Given a sequence x of elements of a commutative equidimensional noetherian ring R  , cycles _zi(x,R)$z_i(\mathbf{x},R)$ (i∈N$i\in \mathbb{N}$) in the cycle group of polynomial rings over R are defined by generic residual intersections. The study of these cycles gives new insight into the theory for excess intersections in projective space developed by Stückrad and Vogel, in particular concerning the contribution to the intersection cycle of embedded components not defined over the base field.     

Isotriviality and étale cohomology of Laurent polynomial rings

We provide a detailed study of torsors over Laurent polynomial rings under the action of an algebraic group. As applications we obtained variations of Raghunathan’s results on torsors over affine space, isotriviality results for reductive group schemes and forms of algebras, and decomposition properties for Azumaya algebras.