This page contains a short resumee of my current researches and arguments that I would like to consider,in some future. My scientific field is ALGEBRAIC GEOMETRY (AMS classification 14xxx) and, more in details, projective varieties in char 0.
For older research project, please refer to the list of papers.

Canonical form of polynomials
I am applying the study of k-secant varieties to spaces of polynomials, in order to determine a description of general homogeneous polynomials in terms of elementary ones (power of linear forms, products, etc.).
Decomposability of tensors
I am applying the study of k-secant varieties to spaces of tensors with fixed rank, in order to collect information on their geometric structure.
Projectively degenerate embeddings
I am studying varieties in projective spaces, whose secant or k-secant or tangent varieties have dimension smaller than the expected ones.
Vector bundles on varieties
I am studying cohomological properties of vector bundles and their Moduli spaces, over surfaces and 3folds of some particular type (abelian, Calabi-Yau, etc.).
Structure of varieties and projective embeddings
I am studying how intrinsic properties of varieties (e.g. the gonality of curves or the speciality of linear series) influence the postulation and the cohomology of the embeddings of these objects in P^r.
Singular curves on surfaces
I am studying the structure of the varieties which parametrizes singular curves on a fixed smooth projective surface. These are relevant for the classification of algebraic varieties and also are connected with some topics in Number Theory (hyperbolic varieties) and Theoretical Physics (quantum cohomology).
 
 
 
 

Canonical form of polynomials

The problem concerns methods for decomposing a homogeneous polynomial, in a canonical way, as a sum of simpler elements (with as few as possible summands). Such a decomposition is classically called canonical form.

Usually, the elementary objects we work with are bounded to a (non-linear) subvariety of the space of polynomial of fixed degree. From this point of view, the problem reduces to the study of k-secant varieties to some special varieties (e.g. Veronese varieties, varieties of decomposable forms, etc.).

The possibility of decomposing a polynomial as a sum of k elementary objects is strictly linked to the (non-)existence of subvarieties, inside Veronese or other special varieties, which are contained in spaces of low dimension, and pass through k general points.

OUTPUTS
(1) Study of complete intersection groups of points that one can find on general hypersurfaces of given degree (joint with E.Carlini and A.V.Geramita - pubbl.n.50 and pubbl.n.56).
(2) A general homogeneous polynomial in three variables can be written as a determinant of a matrix of forms with preassigned degrees (joint with J.Migliore - pubbl.n.62). In 4 variables, a general homogeneous polynomial is a sum of determinants with a low number of summands (joint with A.V.Geramita - pubbl.n.69).
 
 
 
 

Decomposability of tensors

The theory of tensors has several interesting connections with the Algebraic Statistics and with the theory of mixture models. From the point of view of Algebraic Geometry, the main tool consists of the application of results from the theory of secant varieties to the special case of Segre varieties, Grassmann varieties, Veronese varieties, etc.

In particular, through the classical works of Terracini, the study of the dimension and the degree of secant varieties is related to the study of degenerate subvarieties of special varieties. Questions on the structure of spaces of tensors with fixed rank, and their identifiability, translate into questions on which kind of curves, surfaces etc. one can find inside Segre varieties.

It follows that geometrical methods are useful in producing criteria for detecting the identifiability of specific or general tensors, and they yield improvements of criteria based on pure Linear Algebra.

OUTPUTS
(1) For the case of symmetric tensors, geometric methods produce effective criteria for detecting the uniqueness of the decomposition of a given tensor. (joint with E.Ballico - pubbl.n.61 and pubbl.n.66).
(2) For general tensors, it is possible to extend the range in which the identifiability is known to hold, outside of a subset of measure 0 (joint with C.Bocci - pubbl.n.64, joint with G.Ottaviani - pubbl.n.63 and joint with C.Bocci e G.Ottaviani - pubbl.n.68). In particular, for the case of binary tensors, the previous results turn out to be almost sharp.
(3) For linear systems of tensors, one obtains results on generic identifiability by using results on the non-weak-Grassmann defectivity of some Segre variety. (joint with E.Ballico, A.Bernardi and M.V.Catalisano - pubbl.n.65).
(4) Examples of spaces of tensors with fixed rank, such that the identifiability fails for the general point, are provided by the study of degenerate subvarieties (joint with M.Mella and G.Ottaviani - pubbl.n.70).
 
 
 
 

Projectively degenerate embedddings

Varieties emmbedded in projective spaces are endowed with their k-secant varieties, i.e. the closures of the unions of all k-secant spaces. varieties for which some k-secant variety has dimension less than the expected one were called defective and intensively studied by classical italian geometers.

The old point of view was to consider them as algebraic solutions of differential equations. A more modern theory of such varieties was developed by Zak in connection with Hartshorne's conjecture on the existence of varieties with small codimension.

Another natural setting in which these varieties arise is  interpolation in projective spaces: given a finite set of points and multiplicities, study the family of hypersurfaces passing through them with the assigned multiplicities. when all the multiplicities are equal to 2, this amounts to the same as asking about the family of k-tangent hyperplanes in some Veronese embedding of the projective space. This is, in turn, a classical argument and also its extension to general varieties was studied by Severi, Palatini, Terracini, Scorza but only for the initial cases.
We try to extend classical results, going further in the classification of defective surfaces and threefolds.

OUTPUTS:
(1) Classification of weakly defective surfaces, i.e. surfaces such that any k-tangent hyperplane is in fact tangent along a subvariety of positive dimension. This is a necessary step in the classification of defective threefolds. (joint with C.Ciliberto - pubbl.n.36).
(2) Classification of surfaces with defective Grassmann - secants; these are surfaces for which, through a general line contained in some trisecant planes one can draw infinitely many secant trisecant planes. This is the first case of a broad class of degenerate objects. In a spaec of dimension 5, they have no triple points, when projected from a general line. (joint with M.Coppens - pubbl.n.32 and joint with C.Ciliberto - pubbl.n.43).
(3) Study of surfaces which satisfy some differential equation for triple points. These are surfaces such that by imposing one triple point to the intersection with a (tangent) hyperplane, one gets less condition than expected (joint with T.Markwig - pubbl.n.51 and pubbl.n.58).
(4) Classification of defective varieties of dimension 3 (joint with C.Ciliberto - pubbl.n.44).
(5) Study of semi-defective varieties, i.e. varieties such the intersection of their secant spaces has unexpected properties. These varieties satisfy some peculiar differential equations and are oganized in a hierarchy (joint with C.Bocci - pubbl.n.47).
(6) Study of the minimal dimesnion that a k-secant variety can have. The problem is linked with the study of projections of varieties (joint with C.Ciliberto - pubbl.n.57).
(7) Study of varieties whose secant order is bigger than one. That is, for a general point of a k-secant variety one finds a number of secant spaces that is (finite and) bigger than 1 (joint with C.Ciliberto - pubbl.n.49).
(8) Bounds for the regularity of a variety having good general projections to hypersurfaces. (joint with N.Chiarli and S. Greco - pubbl.n.30).
 
 
 
 

Singular curves on surfaces

The classification of algebraic varieties deals with the invariants that we can find on these objects. A general non-sense for contructing such invariants relies in the study of subvarieties that a given variety contains. We get, for instance, in this way the classical concept of the Picard group of divisors, which is connected with the analytic structure of the variety. On a surface S, the Picard group is based on the curves traced on S. A significative study of the invariants should consider their behaviour in a family of varieties: e.g., in the case of surfaces of fixed degree in P³, we know the Picard group of a general surface and we define in this way the Noether-Lefschetz loci, which lead to a more detailed classification.

Recent developments made a step further, considering singular curves inside a class of the Picard group. These researches were motivated both from a Geometric point of view and by some results in theoretical Physics. Varieties parametrizing singular curves inside a fixed linear series of plane curves, the so-called Severi loci, were described quite deeply by recent works of Z.Ran, J.Harris e L.Caporaso. At the same time, the string theory of physicists enlighted the importance of determining rational curves in some particular varieties and introduced the notion of quantum cohomology (see the works of E.Witten, Y.Manin, Konsevitch etc.).

In this setting, H.Clemens, L.Ein e G.Xu showed that general elements of some important families cannot contain curves of low genus. These results are linked to some problems in hyperbolic geometry: a variety X is called hyperbolic when there are no non-constant holomorphic maps from C to X; this notion is relevant for applications to extensions of Mordell problem to varieties of dimension >1. Some papers by Kobayashi, Lang, Demailly suggest that hyperbolicity could be equivalent to have a lower bound,depending on the degree, for the genus of curves in X. Clemens and Xu prove that the genral elements of some natural families have this last property.

In all the previous considerations, it turns out that the Severi loci seem to have analytic, more than algebraic, nature; hence we tried to use some environment from projective differential geometry (namely, the theory of foci together with inductive degeneration) to get informations on them. In particular we pointed our attention to surfaces in P³, for, by now, we should note that the structure of the Severi loci on a general surface in P³ is not understood.

OUTPUTS
(1) A lower bound for the genus of curves on surfaces with Picard group of rank 2 and for the geometric genus of varieties which fill ipersurfaces of any variety. (joint with A.Lopez - pubbl.n.29 and joint with A.Lopez and Z.Ran - pubbl.n.31).
(2) The existence, in some range, of a good component of the Severi locus, i.e. a component which is generically smooth, of the expected dimension, on a general surface (joint with C.Ciliberto - pubbl.n.26).
(3) A sharp limit for the genus, over which the nodal locus, inside the Severi locus, is smooth (joint with E.Sernesi - pubbl.n.23).
(4) The description of sets of points Z in P² whose double imposes few conditions to curves. The problem is strictly linked with the study of symbolic powers of an ideal. (joint with C.Bocci - pubbl.n.59).
 
 
 
 

Vector bundles on varieties

The classification of algebraic varieties deals with the invariants that we can find on these objects. A general non-sense for contructing such invariants relies in the study of subvarieties that a given variety contains. We get, for instance, in this way the classical concept of the Picard group of divisors, which is connected with the analytic structure of the variety.

The Picard group classifies subvarieties of codimension 1. The next step requires a classification of subvarieties of higher codimension. However, yet in codimension 2, the situation is not clear: while the divisors are organized in nice linear series, a similar construction in higher codimension leads to the definition of the Chow ring whose structure is widely unknown.

An alternative arises when we deal with subvarieties of codimension >1 which are 0-loci of sections of vector bundles; in this case, our objects determine elements of some vector spaces (of sections of the vector bundle) which somehow is a substitute for the linear series that we have for divisors. Studying vector bundles and their Moduli we can then derive a deeper description of the underlying algebraic variety. It should be observed that some solutions to some field equation describing interaction of particles in nuclear physics determine algebraic curves associated to vector bundles of some kind (e.g. instantons).

In this topic, several facts are known for rank 2 bundles on surfaces and on P³, but there are few results for bundles on general 3folds and their connection with the Chow ring. One could find here, in my opinion space for some interesting research.

Unifying the two previous arguments, we started the study of varieties parametrizing sections of a vector bundle, whiose 0-loci are singular; by now, we examined the case of rank 2 bundles on P³.

OUTPUTS
(1) Study of bundles without intermediate cohomology (ACM bundles) on hypersurfaces. We extend the Horrocks' decomposability theorem to general sestic 3folds in P⁴ (joint with C. Madonna - pubbl.n. 42) and to hypersurfaces of low degree in P⁵ (joint with C. Madonna - pubbl.n. 45).
(2) Determine the possible Chern classes of ACM bundles on surfaces of low degree in P³. This is linked to the existence of arithmetically Gorenstein groups of points on these surfaces, and also to the pfaffian representation of polynomials (joint with D. Faenzi - pubbl.n.54 and pubbl.n.55 ).
(3) Certain Moduli spaces of rank 2 bundles over a K3 surface S are birational to a symmetric products of S. (joint with E.Ballico - pubbl.n.16).
(4) We outline the theory of Severi varieties for sections of a rank 2 bundle on P³ (joint with E.Ballico - pubbl.n.25).
 
 
 
 

Structure of varieties and projective embeddings

In the theory of algebraic varieties one considers both 'intrinsic' properties, defined up to isomorphism (like geometric genus) and 'extrinsic' properties, related to an embedding in a projective space (e.g the degree). On the other hand, classical geometers yet were aware that the two aspects were not independent and intrinsic properties put conditions on the embeddings; in this contest, Castelnuovo proved the celebrated bound for the genus of a curve in terms of its degree.

Other relevant links between the extrinsic and the intrinsic geometry of curves in P³ were found by Halphen, at the end of the last century, but only recently proved by L.Gruson e C.Peskine (see also the works of J.Harris, R.Hartshorne e A.Hirschowitz): they give bounds for the genus in terms of the degree of surfaces passing through the curve. Tese formulae are generalized in several ways to some particular curves in P^r, but a complete and comprehensive picture is still unknown.

The research consider some aspects of this problem. First of all, there is an attempt to derive a complete generalization of Halphen's theory; furthermore, we observe that deeper intric properties of curves, like e.g. their gonality, imposes conditions on the postulation of the embeddings in a way that is only initially understood; also, the cohomology of linear series on a curve should be considered in this setting. Further developments should consider the extension of these results to higher dimensional varieties and a relative version of the theory, in the framework of Brill-Noether theory.

OUTPUTS
(1) relations among the degree, the genus, the postulation and the gonality of curves in P³ (joint with C.Ciliberto - pubbl.n.24).
(2) relations among the degree, the genus and the flags of subvarieties containing a curve in P^r. The results are particularly sharp in P⁴. (joint with C.Ciliberto and V. di Gennaro - pubbl.n.20 and pubbl.n.22).