Projective modules over smooth, affine varieties over Archimedean real closed fields

Let be a smooth, affine variety of dimension n≥2 over the field R of real numbers. Let P be a projective A-module of such that its nth Chern class is zero. In this set-up, Bhatwadekar-Das-Mandal showed (amongst many other results) that P?A⊕Q in the case that either n is odd or the topological space X(R) of real points of X does not have a compact, connected component. In this paper, we prove that similar results hold for smooth, affine varieties over an Archimedean real closed field .     

On varieties of almost minimal degree III: Tangent spaces and embedding scrolls

Let X⊂P^r be a variety of almost minimal degree which is the projected image of a rational normal scroll from a point p outside of . In this paper we study the tangent spaces at singular points of X and the geometry of the embedding scrolls of X, i.e. the rational normal scrolls Y⊂P^r which contain X as a codimension one subvariety.     

On the variety parameterizing completely decomposable polynomials

The purpose of this paper is to relate the variety parameterizing completely decomposable homogeneous polynomials of degree d in n+1 variables on an algebraically closed field, called , with the Grassmannian of (n−1)-dimensional projective subspaces of P^n+d−1. We compute the dimension of some secant varieties to . Moreover by using an invariant embedding of the Veronese variety into the Plücker space, we are able to compute the intersection of G(n−1,n+d−1) with , some of its secant varieties, the tangential variety and the second osculating space to the Veronese variety.     

Separators of points on algebraic surfaces

For a finite set of points X⊆Pⁿ and for a given point P∈X, the notion of a separator of P in X (a hypersurface containing all the points in X except P) and of the degree of P in X, (the minimum degree of these separators) has been largely studied. In this paper we extend these notions to a set of points X on a projectively normal surface S⊆Pⁿ, considering as separators arithmetically Cohen-Macaulay curves and generalizing the case S=P² in a natural way. We denote the minimum degree of such curves as and we study its relation to . We prove that if S is a variety of minimal degree these two terms are explicitly related by a formula, whereas only an inequality holds for other kinds of surfaces.     

Rational points and cohomology of discriminant varieties

Let G be a finite unitary reflection group acting in a complex vector space . The discriminant varietyX_G of G is defined as the space of regular orbits of G on V. Classical examples include the varieties of complex polynomials of degree n with distinct (resp. non-zero distinct) roots. The normaliser of G in GL(V) acts on X_G; in this work we determine the action of on the cohomology of X_G. In the classical cases this amounts to computing the cohomology of X_G with certain local coefficient systems. Our methods are to compute equivariant weight polynomials by means of explicit counting of the rational points of certain varieties over finite fields, and then to exploit the weight purity of the relevant varieties. We obtain some power series identities as a byproduct.     

Galois closure and Lagrangian varieties

Let X be a complex projective variety and consider the morphism

     

Higher syzygies of hyperelliptic curves

Let X be a hyperelliptic curve of arithmetic genus g and let f:X→P¹ be the hyperelliptic involution map of X. In this paper we study higher syzygies of linearly normal embeddings of X of degree d≤2g. Note that the minimal free resolution of X of degree ≥2g+1 is already completely known. Let A=f^∗O_P¹(1), and let L be a very ample line bundle on X of degree d≤2g. For , we call the pair (m,d−2m)the factorization type ofL. Our main result is that the Hartshorne-Rao module and the graded Betti numbers of the linearly normal curve embedded by |L| are precisely determined by the factorization type of L.     

Projectively simple rings

An infinite-dimensional N-graded k-algebra A is called projectively simple if dim_kA/I<∞ for every nonzero two-sided ideal I⊂A. We show that if a projectively simple ring A is strongly noetherian, is generated in degree 1, and has a point module, then A is equal in large degree to a twisted homogeneous coordinate ring B=B(X,L,σ). Here X is a smooth projective variety, σ is an automorphism of X with no proper σ-invariant subvariety (we call such automorphisms wild), and L is a σ-ample line bundle. We conjecture that if X admits a wild automorphism then every irreducible component of X is an abelian variety. We prove several results in support of this conjecture; in particular, we show that the conjecture is true if . In the case where X is an abelian variety, we describe all wild automorphisms of X . Finally, we show that if A is projectively simple and admits a balanced dualizing complex, then is Cohen-Macaulay and Gorenstein.     

Hilbert curves of polarized varieties

Let X be a normal Gorenstein complex projective variety. We introduce the Hilbert variety V_X associated to the Hilbert polynomial χ(x₁L₁+?+x_ρL_ρ), where L₁,…,L_ρ is a basis of , ρ being the Picard number of X, and x₁,…,x_ρ are complex variables. After studying general properties of V_X we specialize to the Hilbert curve of a polarized variety (X,L), namely the plane curve of degree dim(X) associated to χ(xK_X+yL). Special emphasis is given to the case of polarized threefolds.     

On a question of Sean Keel

Let X be a smooth projective surface defined over , and let L be a line bundle over X such that for every complete curve Y contained in X. A question of Keel asks whether L is ample. If X is a P¹-bundle over a curve, we prove that this question has an affirmative answer.     

A polar de Rham theorem

We prove an analogue of the de Rham theorem for polar homology; that the polar homology HP_q(X) of a smooth projective variety X is isomorphic to its H^n,n−q Dolbeault cohomology group. This analogue can be regarded as a geometric complexification where arbitrary (sub)manifolds are replaced by complex (sub)manifolds and de Rham's operator d is replaced by Dolbeault's .     

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.     

On the fundamental group-scheme

We show that the fundamental group-scheme of a separably rationally connected variety defined over an algebraically closed field is trivial. Let X be a geometrically irreducible smooth projective variety defined over a finite field k admitting a k-rational point. Let {En,σn}_n?0 be a flat principal G-bundle over X, where G is a reductive linear algebraic group defined over k. We show that there is a positive integer a such that the principal G-bundle is isomorphic to E₀, where FX is the absolute Frobenius morphism of X. From this it follows that E₀ is given by a representation of the fundamental group-scheme of X in G.     

The inversion formula for automorphisms of the Weyl algebras and polynomial algebras

Let A_n be the nth Weyl algebra and P_m be a polynomial algebra in m variables over a field K of characteristic zero. The following characterization of the algebras {A_n⊗P_m} is proved: an algebraAadmits a finite setδ₁,…,δ_sof commuting locally nilpotent derivations with generic kernels andiffA?A_n⊗P_mfor somenandmwith2n+m=s, and vice versa. The inversion formula for automorphisms of the algebra A_n⊗P_m (and for ) has been found (giving a new inversion formula even for polynomials). Recall that (see [H. Bass, E.H. Connell, D. Wright, The Jacobian Conjecture: Reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. (New Series) 7 (1982) 287-330]) given, then (the proof is algebro-geometric). We extend this result (using [non-holonomic] D-modules): given, then. Any automorphism is determined by its face polynomials [J.H. McKay, S.S.-S. Wang, On the inversion formula for two polynomials in two variables, J. Pure Appl. Algebra 52 (1988) 102-119], a similar result is proved for .One can amalgamate two old open problems (the Jacobian Conjecture and the Dixmier Problem, see [J. Dixmier, Sur les algèbres de Weyl, Bull. Soc. Math. France 96 (1968) 209-242. [6]] problem 1) into a single question, (JD): is aK-algebra endomorphismσ:A_n⊗P_m→A_n⊗P_man algebra automorphism providedσ(P_m)⊆P_mand? (P_m=K[x₁,…,x_m]). It follows immediately from the inversion formula that this question has an affirmative answer iff both conjectures have (see below) [iff one of the conjectures has a positive answer (as follows from the recent papers [Y. Tsuchimoto, Endomorphisms of Weyl algebra and p-curvatures, Osaka J. Math. 42(2) (2005) 435-452. [10]] and [A. Belov-Kanel, M. Kontsevich, The Jacobian conjecture is stably equivalent to the Dixmier Conjecture. ArXiv:math.RA/0512171. [5]])].     

On the geometry of generalized quadrics

Let {f₀,…,f_n;g₀,…,g_n} be a sequence of homogeneous polynomials in 2n+2 variables with no common zeros in P²ⁿ⁺¹ and suppose that the degrees of the polynomials are such that is a homogeneous polynomial. We shall refer to the hypersurface X defined by Q as a generalized quadric. In this note, we prove that generalized quadrics in for n≥1 are reduced.