Bedlewo School -- Topics

Topics

1. Seshadri constants of ample line bundles
Motivated partly by attempts to prove Fujita's conjecture in higher dimensions, Demailly [Dem92] introduced Seshadri constants which capture the concept of local positivity of an ample line bundle. Roughly speaking, the Seshadri constant of a line bundle L at a point x measures the rate of growth of the number of jets that tensor powers of L generate at x. Whereas originally Seshadri constants were viewed as a useful tool to produce sections of the adjoint line bundle, they quickly gained the interest on their own (see e.g. [Bau99], [Laz97], [Ste98]).
Seshadri constants turned out to be very hard to control, and - apart from abelian surfaces (see [Bau99]) - exact values are known only in very few examples (see [Bau97] and [BauSze01]). Even providing bounds on these numbers is in many cases an interesting but hard problem. Remarkably, the computation of multiple point Seshadri constants on the projective plane is equivalent to the long standing conjecture of Nagata [Nag59]. The discussion of recent developments in this direction should provide the participants with numerous intriguing open questions.

2. Higher order embeddings of algebraic varieties
The notions of k-very ampleness and k-jet ampleness introduced by Beltrametti, Francia and Sommese [BFS] generalize in the natural manner the global generation of a line bundle or its very ampleness. They have been actively studied recently. By now, generation of jets and k-very ampleness are fairly well understood on algebraic surfaces. These notions, as involving the whole geometry of the underlying variety, can be viewed as a way to express the global positivity of a line bundle. Somewhat surprisingly higher order embeddings are related to (and depend on) the local positivity of ample line bundles as described above as well. The lectures will focus on that interplay and on concrete examples, building mainly upon [BDS00], [BDS01], [BauSze97]. This part of the course should end with the conjectural picture of higher order embeddings of higher dimensional varieties.

3. Syzygies of algebraic varieties
The study of linear systems on curves, surfaces and other algebraic varieties has always been central to algebraic geometry. Broadly speaking, the goal is to understand how a variety X can map to projective space. In turn, given a projective embedding X into P_n one needs to have criteria to express how 'good' such an embedding is. Such criteria are provided by the properties N_p, due to Green and Lazarsfeld [Gr2], [GL], telling to which level the successive syzygies are as simple as possible. For instance N₀ means that the embedding is projectively normal, N₁ means that N₀ is satisfied and that the homogenous ideal I of X in P_n is generated by quadrics, and N₂ means that N₀ and N₁ are satisfied and that the relations between the quadratic generators of I are linear with linear polynomials as coefficients. Continuing in this way, this shows that if N_p is satisfied for all p then the embedding is as nice as possible. Green [Gr2] showed that if X is a curve of genus g, embedded into P_n by a complete linear system of degree at least 2g+1+p , then N_p is satisfied. This result let people study analogous questions for varieties of higher dimensions. The purpose of the lectures is to introduce the theory of syzygies and discuss former and recent results concerning the properties N_p.

References

[Bau97] Bauer, T.: Seshadri constants of quartic surfaces, Math. Ann. 309 (1997), 475-481
[Bau99] Bauer, Th., Seshadri constants on algebraic surfaces, Math. Ann. 313, (1999), 547-583
[BDS00] Bauer, Th., DiRocco, S., Szemberg, T.: Generation of jets on K3 surfaces, J. Pure Appl. Alg. 146 (2000), 17-27
[BDS01] Bauer, Th., DiRocco, S., Szemberg, T.: Cyclic coverings and higher order embeddings of algebraic varieties, Trans. Am. Math. Soc. 353 (2001), 877-891
[BauSze97] Bauer, Th., Szemberg, T.: Primitive higher order embeddings of abelian surfaces, Trans. Am. Math. Soc. 349 (1997), 1675-1683.
[BauSze01] Bauer, Th., Szemberg, T.: Local positivity of principally polarized abelian threefolds. J. reine angew. Math. 531 (2001), 191-200
[BFS] Beltrametti, M.C., Francia, P., Sommese, A.J.: On Reider's method and higher order embeddings. Duke Math. J. 58, (1989), 425-439
[Bir95] Birkenhake, Ch.: Linear systems on projective spaces, Manuscripta Math. 88 (1995), 177-184
[Bir96] Birkenhake, Ch.: Noncomplete linear systems on abelian varieties, Trans. Amer. Math. Soc. 348 (1996), 1885-1908
[Dem92] Demailly, J.-P.: Singular Hermitian metrics on positive line bundles. Complex algebraic varieties (Bayreuth, 1990), Lect. Notes Math. 1507, Springer-Verlag, 1992, pp. 87-104.
[EL93] Ein, L., Lazarsfeld, R., Syzygies and Koszul Cohomology of Smooth Projective Varieties of Arbitrary Dimension, Invent. Math. 111 (1993), 51-67.
[Gr1] Green, Mark L., Koszul cohomology and the geometry of projective varieties, J. Differential Geom. 19 (1984), 125-171
[Gr2] Green, M., Koszul Cohomology and the Geometry of Projective Varieties II, J. Differential Geom. 20 (1984),279-289.
[GL] Green, M., Lazarsfeld, R., Some results on the Syzygies of Finite Sets and Algebraic curves. Compositio Math. 67 (1988), 301-314.
[Laz97] Lazarsfeld, R.: Lengths of periods and Seshadri constants of abelian varieties. Math. Res. Letters 3, 439-447 (1997)
[Nag59] Nagata, M., On the 14-th problem of Hilbert, Am. J. Math. 81 (1959), 766-772
[OP01] Ottaviani, G., Paoletti, R., Syzygies of Veronese Embeddings, Compositio Math. 125 (2001), 31-37.
[Par00] Pareschi, G., Syzygies of abelian varieties, J. Amer. Math. Soc. 13 (2000), 651-664
[Ste98] Steffens, A.: Remarks on Seshadri constants. Math. Z. 227, 505-510 (1998)

[Bau97]	Bauer, T.: Seshadri constants of quartic surfaces, Math. Ann. 309 (1997), 475-481
[Bau99]	Bauer, Th., Seshadri constants on algebraic surfaces, Math. Ann. 313, (1999), 547-583
[BDS00]	Bauer, Th., DiRocco, S., Szemberg, T.: Generation of jets on K3 surfaces, J. Pure Appl. Alg. 146 (2000), 17-27
[BDS01]	Bauer, Th., DiRocco, S., Szemberg, T.: Cyclic coverings and higher order embeddings of algebraic varieties, Trans. Am. Math. Soc. 353 (2001), 877-891
[BauSze97]	Bauer, Th., Szemberg, T.: Primitive higher order embeddings of abelian surfaces, Trans. Am. Math. Soc. 349 (1997), 1675-1683.
[BauSze01]	Bauer, Th., Szemberg, T.: Local positivity of principally polarized abelian threefolds. J. reine angew. Math. 531 (2001), 191-200
[BFS]	Beltrametti, M.C., Francia, P., Sommese, A.J.: On Reider's method and higher order embeddings. Duke Math. J. 58, (1989), 425-439
[Bir95]	Birkenhake, Ch.: Linear systems on projective spaces, Manuscripta Math. 88 (1995), 177-184
[Bir96]	Birkenhake, Ch.: Noncomplete linear systems on abelian varieties, Trans. Amer. Math. Soc. 348 (1996), 1885-1908
[Dem92]	Demailly, J.-P.: Singular Hermitian metrics on positive line bundles. Complex algebraic varieties (Bayreuth, 1990), Lect. Notes Math. 1507, Springer-Verlag, 1992, pp. 87-104.
[EL93]	Ein, L., Lazarsfeld, R., Syzygies and Koszul Cohomology of Smooth Projective Varieties of Arbitrary Dimension, Invent. Math. 111 (1993), 51-67.
[Gr1]	Green, Mark L., Koszul cohomology and the geometry of projective varieties, J. Differential Geom. 19 (1984), 125-171
[Gr2]	Green, M., Koszul Cohomology and the Geometry of Projective Varieties II, J. Differential Geom. 20 (1984),279-289.
[GL]	Green, M., Lazarsfeld, R., Some results on the Syzygies of Finite Sets and Algebraic curves. Compositio Math. 67 (1988), 301-314.
[Laz97]	Lazarsfeld, R.: Lengths of periods and Seshadri constants of abelian varieties. Math. Res. Letters 3, 439-447 (1997)
[Nag59]	Nagata, M., On the 14-th problem of Hilbert, Am. J. Math. 81 (1959), 766-772
[OP01]	Ottaviani, G., Paoletti, R., Syzygies of Veronese Embeddings, Compositio Math. 125 (2001), 31-37.
[Par00]	Pareschi, G., Syzygies of abelian varieties, J. Amer. Math. Soc. 13 (2000), 651-664
[Ste98]	Steffens, A.: Remarks on Seshadri constants. Math. Z. 227, 505-510 (1998)