Research activity.

In the last years the research activity of the local group of Catania has been concentrated mostly in the following subjects:

a) Hilbert function of 0-dimensional schemes and curves in projective spaces subjected to some particular geometric conditions; Gorenstein schemes in codimension 3; relations between the postulation of a curve and that of a generic hyperplane section.

b) Vector bundles and linear series on a curve; bounds of Clifford type for stable bundles.

c) Semilocal rings of dimension one and its value semigroup; canonical module and applications to the singularities of curves.

d) Computer algebra; applications of the theory of Groebner bases to differential polynomials and to fractional polynomials.

In these topics various results have been obtained; these results are the contents of some papers already published or to be published.

Program of the local unity for the next years.

The research program that the local unity intends to execute regards the following themes.

A) Hilbert function of 0-dimensional schemes and curves in projective spaces and in particular in P3, subjected to certain geometric conditions; Gorenstein schemes in codimension 3; relations between the postulation a curve and that of a generic hyperplane section. In particular we intend to give an answer to the following question:
- A1) Given a 0-dimensional Gorenstein subscheme of P3, contained in a smooth surface, if it is the intersection of two ACM curves.
- A2) Given a smooth curve in P3, give a bound on the minimal degree of a surface containing it, in terms of the postulation of a generic plane section.

B) Vector bundles and linear series on curves.
- B1) We intend to characterize families of curves which are special with respect to Brill-Noether theory of rank >1, i.e. curves with bundles of rank >1, with many global sections.
- B2) We intend to study, in characteristic p, the curves, and more generally the varieties, in terms of the rank of Cartier operator; this will be useful in order to give a stratification of the moduli space in terms of this invariant.

C) Semilocal domains of dimension 1 and their value group. The canonical module and application to the singularities of curves. In particular the following problems will be faced: to characterize the semigroups which are the value semigroup of some ring; to classify the singularities of a curve in terms of this semigroup.

D) Computer algebra, Groebner bases and application. We intend to apply the theory of Groebner bases to differential polynomials and to the elimination theory in differential algebra. This would give computational methods that can be applied to differential equations.

List of the main papers of the local unity in the last few years.

G.Carrà-Ferro: -Systems of Non homogeneous Linear Partial Differential Equations and Differential Resultants
Modern Group Analysis VI: Developments in Theory, Computation and Application, New Age International Publishers (1996), 401-409

G.Carrà-Ferro:-A Resultant Theory for the Systems of Two Ordinary Algebraic Differential Equations Applicable Algebra in Engineering, Communication and Computing, Springer-Verlag vol.8 (1997), 539-560

G.Carrà-Ferro:-Differential Grobner Bases in One Variable and in the Partial Case Mathl. Comput. Modelling, Pergamon Press vol.25 (1997), 1-10

G.Carrà-Ferro: -A Resultant Theory for Ordinary Algebraic Differential Equations Procedings AAECC-12, Lect. Notes in Comp. Sci., Springer vol.1255 (1997), 55-65

G.Carrà-Ferro, G.Gallo and R.Gennaro:-Probabilistic Verification of Elementary Geometry Statements Automated Deduction in Geometry, Int. Workshop in Toulouse, 1996 (D.Wang ed.), Lect. Notes in Artificial Intelligence, Springer vol.1360 (1998), 87-101

G.Carrà-Ferro: -Triangular Matrices, Differential Resultants and Systems of Linear Homogeneous PDE's Mathematics and Computers in Simulation, North-Holland (1998), 511-518

G.Carrà-Ferro: -Computer Algebra, Nonlinear Differential Equations and Analog Circuit Design Proc. Int. Workshop Symb. Methods Appl. Circuit Des. SMACD'98, ITW Kaiserslautern (1998), 130-133

A. Causa: -Sulle curve di genere massimo Le Matematiche Vol.53 (1998) Fascicolo 1, pp. 155-175

A. Causa, R. Re, T. Teodorescu:.-Some remarks on linear spaces of nilpotent matrices Le Matematiche, special issue PRAGMATIC 1997 Vol LIII (1998)

M. D'Anna:- Type Sequences of Numerical Semigroups . Semigroup Forum, 56 (1998), 1-31.

M. D'Anna:- Canonical module and one-dimensional analytically irreducible Arf domains.Proceedings of the Fes Conference on Commutative Ring Theory Dekker Lecture Notes in Pure and Applied Mathematics 185 (1997).

M. D'Anna:- The canonical module of a one-dimensional reduced ring . Communications in Algebra, 25 (1997), 2939-2965.

V.Barucci, M.D'Anna, R.Froberg: - The semigroup of values of a one-dimensional local ring with two minimal primes . (1998) in print on Communications in Algebra

V.Barucci, M.D'Anna, R.Froberg: - Analytically unramified one-dimensional semilocal rings and their value semigroup . (1998) in print on Journal of Pure and Appl. Algebra

M.D'Anna, D.Delfin -Integrally closed ideals and type sequences in one dimensional local rings .(1998) accettato su Rocky Mountain J. Math.

S.Giuffrida,R.Maggioni,A.Ragusa:- Resolutions of generic points lying on a smooth quadric Manu. Math.91, 421-444 (1996)

R.Maggioni, A.Ragusa- Betti numbers of space curves bounded by Hilbert functions Le Matematiche LII (1997) 217-232

G.Paxia,A.Ragusa - Generic quadric sections of irreducible Buchsbaum curves J.Pure and Appl. Algebra 119 (1997)

G. Raciti: - Sulla funzione di Hilbert di sottoschemi zerodimensionali di P3 contenuti in una superficie Cubica Sem. Mat.Univ.Pol. Torino Vol.55 (1997) 203-211

A.Ragusa,G.Zappalà - Postulation of subschemes of irreducible curves on a quadric surface In print on Rend.Circ.Matem. Palermo (1999)

A.Ragusa,G.Zappalà -Properties of 3-codimensional Gorenstein schemes Preprint (1999)

R.Re:- Multiplication of sections and Clifford bounds for stable bundles on a curve. Comm. in Algebra vol 26, no 6, (1998)

R.Re: The rank of the Cartier operator and linear systems on curves .Submitted to Journal of Algebra (1998)

R.Strano:- Curves and their hyperplane sections in corso di stampa (special volume dedecated to D. Buchsbaum) (1998)

G. Zappalà:- 0-dimensional subschemes of curves lying on a smooth quadric surface Le Matematiche VOL.LII (1997), 115-127