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# Algebra and Algebraic Geometry

## Team and Interlocutors

A. Garcia, A. Hefez, A. Simis, K. O. Stöhr, I. Vaisencher and their collaborators especially at IMPA, UF. Pernambuco, UF. Fluminense.

S. Kleiman (MIT), H. Stichtenoth (Essen)

## Linear Systems on Gorenstein Curves

There are a lot of publications on linear systems on non-singular curves, and many of them extend naturally to Gorenstein Curves. However there are phenomena, which can only occur on singular curves. A linear system may have a base point, that cannot be removed. Such a point is necessarily a singular point. Stöhr intends to analyze linear systems locally near a base point, to classify Gorenstein curves equipped with 1-dimensional linear systems, and to study their moduli variety.

## Derivations and Bounds for the Degrees of Vector Fields Tangent to a Variety

Let N be a projective variety and let T_Ñ denote the module of tangent vector fields to the affine cone Ñ over N, i.e., the submodule of the module of vector fields of affine space A_k^(n+1) that keep Ñ invariant. This is a finitely generated graded R-module, where R is a polynomial ring over k in n+1 indeterminates. The ultimate goal of this research is a reasonable description of the typical invariant of this module in terms of the given V. A more immediate, perhaps less ambitious, objective is to find sharp bounds for the degrees of a minimal set of generators of T_Ñ away from the trivial vector fields whose that vanish everywhere on N). This problem has been studied in the realm of complex analytic vector fields. The case of a smooth hypersurface has been explicitly obtained by M. Soares (Federal University of Minas Gerais, Brazil); later, an easy proof has been found by E. Esteves (IMPA).

As pointed out by Simis, the idea of the proof was available quite earlier in a paper by J. Lipman, inspired by suggestions of O. Zariski. In fact, quite independently, this researcher has considered earlier the general case of a smooth ideal-theoretic complete intersection by looking at a suitable complex of modules. Stimulated by the netly algebraic contents of the question, Esteves and this researcher are working on obtaining the free R-resolution of the module Im(I) + R^n+1, from which the sought degrees can be read off the first syzygy module of the latter. (Here, (I) denotes the jacobian matrix of a set of minimal homogeneous generators of I).

## Algebraic Theory of Singularities

The algebraic theory of singularities was essentially founded by Oscar Zariski in the 60's and beginning of the 70's who dedicated to it several famous works. A central and still open problem in this theory is the analytic classification of germs of analytic singular plane curves and the construction of the corresponding Moduli spaces.

A. Hefez been working on this question and progressed significantly in the direction of determining numerical invariant that could make possible such classification. These invariant are obtained by studying the module of Kahler differentials of the ring of the singular curve and computed by means of algorithms we developed jointly with our former Ph.D. student M. Hernandes. Besides allowing to understand deformations of structures of singularities of curves, our study may be useful in enumerative geometry, where recently S. Kleiman e R. Piene made use of the theory of plane curves singularities to state and solve enumerative problems involving curves with prescribed singularities. This opens the possibility of renewing our interaction with these researchers. Other themes Hefez is investigating, on one hand, the study of rings and modules of differential operators on the local ring of a singular curve, where we intend to relate properties of the operators with properties of the singularities. This is a joint project with Daniel Levcovitz. On the other hand, jointly with N. Kakuta he is investigating the properties of families of plane curve singularities in positive characteristic where, for example, he is interested in studying the validity or not of results of the type " constant implies topological triviality", of L. Ramanujan. This will imply the complete algebrization of the theory which in the actual stage uses heavily topological and transcendental methods.

## Enumerative Properties of Families of Curves

I. Vainsencher and L. Gatto are interested in enumerative properties of families of curves. There are two variations. The first one deals with M_g, the moduli space of curves of genus g or a suitable compactification thereof. In this case, our main goal is to understand the enumerative geometry of some natural cycles, such as defined by hyperellipticity and/or imposition of special Weirstrass points. The problem of finding divisors classes in M_g corresponding to curves having some special property (e.g. special linear series, theta characteristics,...) still needs some investigations. For instance we wish to determine a Porteous Formula with Excess suitable to situations where a map of vector bundles degenerates along subvarieties in the "wrong codimension". We are already working out some significant examples. Such a formula may be applied in a great variety of situations, such as "of the moduli space corresponding to curves having special Weierstrass points", "relation in the tautological ring of M_g, problems related with the Brill-Nother theory, studying classes of varieties of special divisors on curves not with general moduli" (problem suggested by C. Ciliberto). We have already some results (joint with Eduardo Esteves "A new proof and a geometric interpretation of a relation by Cornalba and Harris", submitted; "Families of Wronskian Correspondences" appeared in "Geometric and Combinatorial aspects of commutative algebra" (Herzog, Restuccia, eds.) 2001, and "Limits of special Weierstrass points", with C. Cumino, preprint 2001).

The second variation has to do with the "situation pre-passage to the quotient". We mean by this the study of the full component Hilbert schemes of a few curves of low genus and degree, suitable for enumerative applications via Bott's residue formula. We hope to be able to determine some characteristic numbers, avoiding Gromov-Witten subtleties that involve virtual numbers the enumerative significance of which is unclear.

## Genera of Curves and Error Correcting Codes

A celebrated theorem of A. Weil gives an upper bound for the number of rational points on curves over finite fields. This bound is given in terms of the cardinality of the finite field and of the genus of the curve. This theorem is equivalent with the validity of the Riemann Hypothesis for the associated zeta-function, which was introduced by E. Artin.

The interest on algebraic curves over finite fields with many rational points was renewed after the introduction by V. D. Goppa of the so-called Geometric Codes (also called Goppa Codes). An application with a tremendous impact among specialists (especially among the engineers in Information Theory) was the construction due to Tsfasman-Vladut-Zink of an infinite sequence of linear codes with limit parameters above the asymptotic Gilbert-Varshamov bound. When the number of rational points of an algebraic curve reaches the upper bound of the theorem of A. Weil ,the curve is called maximal. A result of Y. Ihara bounds the genus of a maximal curve in terms of the cardinality of the finite field. Hence we have two natural problems to consider(over a fixed finite field):

• Genus spectrum : Consists in determining the possible genera of maximal curves.
• Classification: Consists in determining all maximal curves of a given genus.

In these problems A. Garcia has been interacting with H. Stichtenoth, F. Torres, G. Korchmaros and C. P. Xing.

Another research direction is the development of effective methods for the construction of algebraic curves with many rational points over finite fields. Many rational points meaning a number close to the best known upper bound for curves with the same genus. Here the interaction has been with G. van der Geer, L. Quoos and A. Garzon.

Another main subject of Garcia's recent mathematical research: The construction of an explicit infinite sequence of algebraic curves (with increasing genera) having a positive limit for the infinite sequence of ratios of number of rational points over the genus. This limit of ratios is bounded from above by a result of Drinfeld-Vladut. When the cardinality of the finite field is a square, it is a result of Y. Ihara that the bound of Drinfeld-Vladut is attained. This result is central for the construction due to Tsfasman-Vladut-Zink mentioned earlier. For its effective use in Coding Theory, it is though fundamental that the curves in the infinite sequence are explicit given by algebraic equations and that the coordinates of the rational points are also explicitly given. This is developed jointly with H. Stichtenoth.