School on "Liaison and related topics"
Politecnico Torino 1-5 October 2001.
The school has been organized by G. Casnati, N. Chiarli and S. Greco.
The main lecturers were: J. Migliore (Notre Dame Univ. USA) and U. Nagel (Paderborn Univ. Germany).
Each lecturer has given five lectures, covering the following arguments:
Background: Hyperplanes sections. Deficiency modules. All modules occur (up to shift). Shifts, minimal shifts. Basic double links (as ideals); all shifts occur. Miimal free resolutions.
Gorenstein ideals: Different constructions: BR sheaves, sums of linked ideals, BE in codim 3, K+H, includes CI in general, equals CI in codimension two, examples.
Basic liaisons definitions: First results: standard exact sequences, degrees, arithmetic genera, linear equivalence, basic double links, basic double G-links, applications, another construction of Gorenstein ideals.
More refined results: N-type resolution, E-type resolution, all possibilities occur, Rao's correspondance, Hartshorne-Schenzel theorem. Different notions of minimality. Examples in projective 4-space. All complete intersections are linked. Definition of licci and glicci.
Codimension two results: Hilbert-Burch. Gaeta's theorem. Rao's theorems. Lazarsfeld-Rao (LR) original theorem, LR property. Applications. Vector bundles. Stick figures. Rao's correpondance in higher codimension. In particular converse of Gaeta's theorem and of Hartshorne-Schenzel's theorem for curves.
Some evidence: Gaeta theorem for determinantal ideals. Monomial ideals result. ACM curves on general smooth rational surfaces. Results on scrolls (Hartshorne, Lesperance, Casanellas-Miro Roig, Nagel-Notari-Spreafico). Are all Gorenstein ideals glicci? Other invariants for CI-liaison. Contrast CI-liaison and G-liaison.
Further applications: Monomial ideal curves. Maximal Betti numbers for reduced Gorenstein schemes.
Participants.
Bocci C. (Torino Univ.), Casanellas M. (Barcellona ES), Casnati, G. (Pol. Torino), Chiarli N. (Pol. Torino), Di Gennaro R. (Napoli Univ.), Di Nardo R. (Torino Univ.), Failla G. (Messina Univ.), Ferrara M.R. (Roma III Univ.), Fontanari C. (Pisa SNS), Fumasoli S. (Zurigo CH), Greco S. (Pol. Torino), Migliore J.C. (Notre Dame Univ. USA), Molica Bisci G. (Messina Univ.), Nagel U. (Paderborn Univ. D), Notari R. (Pol. Torino), Roggero M. (Torino Univ.), Sabadini I. (Pol. Milano), Sanchini P. (Brescia-Cattolica),Schlesinger E. (Pol. Milano), Spreafico M.L. (Pol. Torino), Tonoli F. (Goettingen D), Zappalą G. (Catania Univ.)
Comunications.
On 10/3/01 and on 10/4/2001, in the afternoon, there have been some short comunications:
Schlesinger, E.: "Hilbert schemes of curves of degree 4"
Sabadini, I.: "Hilbert scheme of curves of degree d and genus (d-3)(d-4)/2 -1"
Bocci, C.: "Construction of Gorenstein ideals and applications"
Ferraro, M.R.: "Linkage by complete intersections on singular rational normal surfaces and 3-folds"
Di Gennaro, R.: "On the Rao module of curves on rational normal scrolls".
It is planned to publish the proceedings of the school as a special issue of Rendiconti del Seminario dell'Universitą e Politecnico di Torino.