# Institute of Mathematics of the Romanian Academy

## European algebraic geometry research training network

### Cristian Anghel

Cristian.Anghel@imar.ro

Institute of Mathematics of the Romanian Academy
PO Box 1-764, RO 70700 Bucharest, Romania

Scientific proposal:

My principal interest concerns actually the generalizations of the classical Maslov index as secondary characteristic classes, with applications to spontaneous symetry breaking of Yang-Mills fields following the dimensional reduction mechanism. In algebraic-geometric terms, certain homogeneous stable vector bundles on an algebraic variety correspond after dimensional reduction, using the results of Garcia-Prado and Bradlow, to a stable pair on the quotient variety. The geometric quantization problem that corresponds to the associated Yang-Mills-Higgs system depends on the higher dimensional Maslov classes and the problem is to find the relations which are between these naturally related structures.

Publications:

• La stabilité de la restriction à une courbe lisse d'un fibré de rang 2 sur une surface algébrique, Math. Ann. 304 (1996), 53-62.
• Fibrés vectoriels stables avec \chi=0 sur une surface abelienne simple, to appear in Math. Ann. 721 (1999).

### Marian Aprodu

Marian.Aprodu@imar.ro

Institute of Mathematics of the Romanian Academy
PO Box 1-764, RO 70700 Bucharest, Romania

Scientific proposal:

The aim of this research plan is an investigation into the projective geometry of the moduli spaces of rank-2 vector bundles. We seek especially results on their ideals in certain known projective embeddings, as well as the geometry of their Brill-Noether subschemes. The principal case we propose to study is the moduli space {\cal SU}_C(2) of rank 2 vector bundles with trivial determinant, embedded in the 2\Theta-linear system. It is a locally factorial, irreducible, projective variety. Its non-stable part can be canonically identified to the Kummer variety Km(J) of J (if g \geq 3, Km(J) is precisely the singular locus of {\cal SU}_C(2)). Its Picard group is generated by a line bundle {\cal L}, which gives rise to a finite morphism \theta:{\cal SU}_C(2) \rightarrow |{\cal L}|^* \cong |2\Theta|.

There are very important things known about the structure of the morphism \theta. For example, if C has genus two, then \theta is an isomorphism of {\cal SU}_C(2) onto |2\Theta|\cong{\bf P}^3, if C is hyperelliptic of highere genus, then \theta is 2-to-1 onto a subvariety of |2\Theta| that can be described in an explicit way, and if C is not hyperelliptic of higher genus, \theta is of degree 1 onto its image.

If C has genus three, a result of Narasimhan and Ramanan says that \theta is an isomorphism to a special Heisenberg-invariant quartic Q_C\subset |2\Theta| \cong {\bf P}^7, called the Coble quartic of C. It is characterised by either of two properties: Q_C is the unique Heisenberg-invariant quartic containing Km(J) in its singular locus; and Q_C is the set of 2\Theta-divisors containig some translate of the image of the Abel-Jacobi map W_1 \subset J^1(C). In the genus four case, assuming that C has no vanishing theta-nulls, Oxbury and Pauly got two distinct hypersurfaces in |2\Theta| \cong {\bf P}^{15}: a unique Heisenberg-invariant quartic Q_C \subset |2\Theta| containing the image of {\cal SU}_C(2) in its singular locus, and G_C \subset |2\Theta|, the set of 2\Theta-divisors containig some translate of W_1 \subset J^1(C).

For higher-genus curves there is no similar result known up to now. It seems that one needs firstly to get a better understanding of the genus four case. There are several basic open questions about the hypersurfaces G_C and Q_C, which we propose to answer to. For example, is G_C a quartic? We could also try to deal with a converse problem: instead of asking whether such a given hypersurface has degree four, we can ask how can one construct quartics singular along {\cal SU}_C(2), and thus singular along Km(J) (Narasimhan and Ramanan conjectured that the moduli space is set-theoretically cut out by quartics, so one expects plenty)? It is possible G_C and Q_C to be elements in a special family of Heisenberg-invariant quartics, and it would be nice to find out what kind of canonical special families, containing Q_C and G_C among their members, can appear. We also propose to decide whether the singular locus of Q_C and the image of {\cal SU}_C(2) in |2\Theta| are coincident or not.

Collaboration:

Dr. William Oxbury, Dept. Mathematical Sciences, University of Durham, United Kingdom.

Publications:

• Moduli spaces of vector bundles over ruled surfaces, Nagoya Math. J., vol. 149 (1999) (with V. Brînzanescu)
• Stable rank-2 vector bundles over ruled surfaces, C.R. Acad. Sci. Paris 325, Série I, (1997) (with V. Brînzanescu)

### Nicusor Dan

Nicusor.Dan@imar.ro

Institute of Mathematics of the Romanian Academy
PO Box 1-764, RO 70700 Bucharest, Romania

Scientific proposal:

Arakelov geometry, Green currents.

Publications:

• Prolongement méromorphe des courants de Green, Math. Ann. 1999.
• Fonctions zeta d'Igusa et hauteurs des hypersurfaces, Ann. Polon. Math. LXXI.1 (1999).

### Marius Marchitan

mmarchitan@yahoo.com, marius@lsm.usv.ro,

Department of Sciences, Suceava University
Suceava, Romania

Scientific proposal:

The study of moduli spaces of stable vector bundles over algebraic surfaces.

Publications:

None, new doctorand.

### Mihai-Sorin Stupariu

stupariu@pcnet.ro

Universitatea Bucuresti, Facultatea de Matematica
str. Academiei, nr. 14, 70109 Bucuresti

Scientific proposal:

Stability concepts for oriented holomorphic pairs coupled with Higgs fields; vortex equations on Hermitian manifolds

Publications:

• The Kobayashi-Hitchin correspondence for vortex-type equations coupled with Higgs fields, PhD thesis, Zürich, 1998
• Dimensional reduction of monopole equations, preprint, Bucharest, 2000

Institute of Mathematics of the Romanian Academy
PO Box 1-764, RO 70700 Bucharest, Romania

Scientific proposal:

Special rational curves (e.g. almost-lines, quasi-lines on complex projective varieties), formal geometry and projective geometry.

Former collaborations:

M. Beltrametti, M. Schneider; lecture courses for doctorands at the Univ. Milano.

Publications:

• Algebraic Barth-Lefschetz theorems, Nagoya Math.J. 142 (1996), 17--38.
• Almost-lines and quasi-lines on projective manifolds (with M. Beltrametti and P. Ionescu). Proc. De Gruyter 1999.

### Vasile Brînzanescu

Vasile.Brinzanescu@imar.ro

Institute of Mathematics of the Romanian Academy
PO Box 1-764, RO 70700 Bucharest, Romania

Scientific proposal:

The existence of holomorphic structures in topological vector bundles over compact complex surfaces, the existence of stable vector bundles over some classes of algebraic surfaces and the study of the moduli spaces of vector bundles over surfaces. Collaboration with G. Trautmann, Kaiserslautern University.

Publications:

• Holomorphic Vector Bundles over Compact Complex Surfaces, Lect. Notes in Math. 1624, Springer 1996.
• Moduli spaces of vector bundles over ruled surfaces, Nagoya Math. J., vol. 149 (1999). (with M. Aprodu)

### Nicolae Buruiana

Nicolae.Buruiana@imar.ro

Institute of Mathematics of the Romanian Academy
PO Box 1-764, RO 70700 Bucharest, Romania

Scientific proposal:

Intersection theory on abelian varieties, Albanese varieties, Jacobians and symmetric products.

Publications:

• Singular projective threefolds whose hyperplane sections are ruled non-rational surfaces, Revue Roum. Math. Pures Appl., vol. XXXV,no 8--10 (1990).
• Algebraic cycles on symmetric products and abelian varieties, to appear in Revue Roum. Math. Pures Appl. 1999.

### Gabriel Chiriacescu

Gabi.Chiriacescu@imar.ro

Institute of Mathematics of the Romanian Academy
PO Box 1-764, RO 70700 Bucharest, Romania

Scientific proposal:

Finiteness of Bass numbers of local cohomology modules. Collaborations with: M. Brodmann, Zurich Univ. and R.Y. Sharp, Shefield Univ.

Publications:

• Cofiniteness of local cohomology modules over regular local rings, to appear in Bull. London Math. Soc. (1999).
• A formal complex of Cech type with an application to Hartshorne-Lichtenbaum vanishing theorem, to appear in Revue Roum. Math. Pures Appl. (2000).

### Iustin Coanda

Iustin.Coanda@imar.ro

Institute of Mathematics of the Romanian Academy
PO Box 1-764, RO 70700 Bucharest, Romania

Scientific proposal:

General plane sections of space curves in Range B (via the method of Strano, generic initial ideals and generic initial cohomology modules). Collaboration with G. Floystad, Bergen Univ.

Publications:

• On Barth's restriction theorem, J. Reine Angew. Math., 428 (1992), 97-110.
• Restriction theorems for vector bundles via the methods of Roth and Strano, J. Reine Angew. Math., 487 (1997), 1-25.

### Nicolae Manolache

Nicolae.Manolache@imar.ro

Institute of Mathematics of the Romanian Academy
PO Box 1-764, RO 70700 Bucharest, Romania

Scientific proposal:

The use of syzygies in the study of Hibert schemes or moduli spaces. The first study which I would like to accomplish is that of certain Hilbert schemes of curves in P^3, beginning with the Hilbert schemes of degree 4 Cohen Macaulay curves. One of the difficult problem is the decomposition in connected or irreductible components. For degree up to 3 there is known in the literature that the Hilbert scheme is connected. The difficulty lies in the understandig of the curves with nilpotents. Possible collaboration with W. Decker, F.O. Schreyer and may be also other mathematicians.

Publications:

• Geometry of the Horrocks bundle on P^5, (Proc. of NAVF Conf. in Alg. Geom. Bergen 1989), London Math. Soc. Lecture Notes 179, Complex Geometry 1992, Cambridge University Press (with W. Decker, F.O. Schreyer)
• Moduli of (1,7)-polarized abelian surfaces via syzygies, Duke eprint GA 9812271 (with F.O. Schreyer)

### Ovidiu Pasarescu

Ovidiu.Pasarescu@imar.ro

Institute of Mathematics of the Romanian Academy
PO Box 1-764, RO 70700 Bucharest, Romania

Scientific proposal:

Topics concerning the classification of projective embedded curves.

Let's denote by I'(d,g,n) (resp. I"(d,g,n)) the union of components of the Hilbert scheme H(d,g,n) corresponding to smooth, irreducible, non-degenerate (resp. linearly normal) curves in P^n.
P1: Find the integers (d,g,n), n > 2, such that I'(d,g,n) is nonempty.
P2: Find the integers (d,g,n), n > 2, such that I"(d,g,n) is nonempty.
P3: For each triplet such that I'(d,g,n) is nonempty, try to give its description ("good" components, "bad" components, irreducibility, ...).

Collaborations:

I had collaborations with A. Lascu, Ph. Ellia (Ferrara), A. Hirschowitz (Nice), D. Laksov (Stokholm), J. Alexander (Angers), R. Strano (Catania) on the Problems P1 and P3. I have in present a project with J. Kleppe (Oslo) on P2.

Proposed new (or continuation of old) collaborations with:

A. Hirschowitz (Nice), J. Kleppe (Oslo), Rosa Maria Miro-Roig (Barcelona), Ph. Ellia (Ferara), J. Alexander (Nice).

I need short-term visits (1-2 weeks). If possible, I need a longer collaboration with A. Hirschowitz.

Publications:

• Discrete and continous invariants and classification problems in algebraic geometry, thesis, Bucharest, 1999. (chapter II: Halphen-Castelnuovo theory for curves in P^n, n > 7; chapter III: Linearly normal curves in P^n, n = 4, 5 or n > 7).

### Dorin Popescu

Dorin.Popescu@imar.ro

Institute of Mathematics of the Romanian Academy
PO Box 1-764, RO 70700 Bucharest, Romania

Scientific proposal:

Maximal Cohen-Macaulay modules over isolated hypersurface singularities and combinatorics in Commutative Algebra and Algebraic Geometry. Collaborations with J. Herzog (Essen Univ.), G. Pfister (Kaiserslautern Univ.), M. Roczen (Humboldt Univ. Berlin), B. Martin (Cottbus Univ.), L. O'Carroll (Edinburgh Univ.).

Publications:

• Thom-Sebastiani problems for maximal Cohen-Macaulay modules, Math. Ann. 309 (1997), 677-700 (with J. Herzog)
• A family of Cohen-Macaulay modules over singularities of type X^t+Y^3, Comm. Alg. 1999 (with G. Pfister)

### Andrei Teleman

ateleman@geo.math.unibuc.ro

Department of Mathematics, Bucharest University,
Str. Academiei 14, Bucharest, Romania

Scientific proposal:

Non-abelian Seiberg-Witten theory.

Former collaborations:

With M. Luebke, Holomorphic bundles on non-Kähler manifolds;
With C. Okonek, Seiberg-Witten theory;
With C. Okonek, A. Schmitt, Invariant theory.

Proposed collaboration:

M. Luebke, C. Okonek, Universal Kobayashi-Hitchin correspondence.

Publications:

• Master spaces and the coupling principle: From Geometric Invariant Theory to Gauge Theory, to appear in Comm. Math. Phys. (1999) (with C. Okonek)
• Fredholm L^p-theory for coupled Dirac operators on the Euclidian space, to appear in C.R. Acad. Sci. Paris (1999).

### Cristian Voica

voica@al.math.unibuc.ro

Department of Mathematics, Bucharest University,
Str. Academiei 14, Bucharest, Romania

Scientific proposal:

The classification of quasi-linearly connected manifolds.

Publications:

• Classifications of projective varieties, thesis, Bucharest 1999.
• On a contractibility criterion of curves on projective surfaces, to appear in Revue Roum. Math. Pures Appl. 1999.

### Victor Vuletescu

vvuletes@geo.math.unibuc.ro

Department of Mathematics, Bucharest University,
Str. Academiei 14, Bucharest, Romania

Scientific proposal:

Holomorphic vector bundles on non-projective surfaces, description of some classes of vector bundles (filtrable, simple, stable). Subvarieties of codimension two in projective spaces: surfaces in P^4, classification of surfaces of small degree, syzygies, 3-folds in P^5. Collaboration with F.X. Gallego, Spain.

Publications:

• Vector bundles with zero Chern numbers on elliptic surfaces, to appear in Rendiconti Univ. Trieste (1999).
• A simple proof of Banica-Le Potier's inequality \Delta \geq 0, Ann. Univ. Bucuresti (1999).

Vasile.Brinzanescu@imar.ro