Speaker: Rosa Miro-Roi (University of Barcelona)
Time: 9:00 - 10:30 or 11:00
Venue: Room 6, Building A14, 18 Hoang Quoc Viet Cau Giay, Ha noi
Abstract: In my lectures, I will construct families of non-isomorphic Arithmetically Cohen-Macaulay (ACM) sheaves (i.e. sheaves without intermediate cohomology) on a projective variety X. When X is ACM, in terms of the associated R_X-module E = H^0_*(E) := oplus_t H^0(X, E(t)), they correspond to Maximally Cohen-Macaulay (MCM) modules, i.e., modules that verify depth E = dim R_X. This correspondence allows us to study the problem alternatively from the algebraic or the geometric point of view. The study of such vector bundles (or modules) has a long and interesting history behind. Since the seminal result by Horrocks characterizing ACM bundles on P^n as those that split into a sum of line bundles, an important amount of research has been devoted to the study of ACM on a given variety.
ACM sheaves also provide a criterium to determine the complexity of the underlying variety. More concretely, this complexity can be studied in terms of the dimension and number of families of indecomposable ACM sheaves that it supports, namely, its representation type. Along this lines, varieties that admit only a finite number of indecomposable ACM sheaves (up to twist and isomorphism) are called of emph{finite representation type}. These varieties are completely classified: They are either three or less reduced points in P^2, a projective space P^n, a smooth quadric hypersurface X subset P^n , a cubic scroll in P^4 , the Veronese surface in P^5 or a rational normal curve.
On the other extreme of complexity we would find the varieties of representation type, namely, varieties for which there exist r-dimensional families of non-isomorphic indecomposable ACM sheaves for arbitrary large r. In the case of dimension one, it is known that curves of wild representation type are exactly those of genus larger or equal than two. In dimension greater or equal than two few examples are know and in my talk, I will give a brief account of the known results. More precisely,
In Lecture 1, I will solve a long-standing problem in Algebraic Geometry/Commutative Algebra: I will determine the graded Betti numbers of a minimal free resolution of a general set of points on a del Pezzo surface. In Lecture 2 and as an application of the results obtained in the first lecture, I will determine the representation type of any smooth del Pezzo surface (i.e. the behavior of the associated category of undecomposable ACM bundles). In addition, I will also determine the representation type of Segre varieties. Finally, in Lecture 3, I will analyze how the representation type of a projective variety change when we change the polarization (i.e. the embedding).
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