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Lectures on the Representation Type of a Projective Variety

  • Rosa M. Miró-RoigEmail author
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2210)

Abstract

In these notes, we construct families of non-isomorphic Arithmetically Cohen Macaulay (ACM for short) sheaves (i.e., sheaves without intermediate cohomology) on a projective variety X. The study of such sheaves has a long and interesting history behind. Since the seminal result by Horrocks characterizing ACM sheaves on \(\mathbb {P} ^n\) as those that split into a sum of line bundles, an important amount of research has been devoted to the study of ACM sheaves on a given variety.

ACM sheaves also provide a criterium to determine the complexity of the underlying variety. This complexity is studied in terms of the dimension and number of families of undecomposable ACM sheaves that it supports, namely, its representation type. Varieties that admit only a finite number of undecomposable ACM sheaves (up to twist and isomorphism) are called of finite representation type. These varieties are completely classified: They are either three or less reduced points in \({\mathbb {P}}^2\), \({\mathbb {P}}^n\), a smooth hyperquadric \(X\subset {\mathbb {P}}^n\), a cubic scroll in \({\mathbb {P}}^4\), the Veronese surface in \({\mathbb {P}}^5\) or a rational normal curve.

On the other extreme of complexity we find the varieties of wild representation type, namely, varieties for which there exist r-dimensional families of non-isomorphic undecomposable 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 these notes, we give a brief account of the known results.

Notes

Acknowledgements

The author is grateful to Professor Le tuan Hoa and to Professor Ngo Viet Trung for giving her the opportunity to speak about one of her favorite subjects: Arithmetically Cohen-Macaulay bundles on projective varieties and their algebraic counterpart Maximal Cohen-Macaulay modules. She is also grateful to the participants for their kind hospitality and mathematical discussions that made for a very interesting and productive month in the lovely city of Hanoi.

Last but not least, I am greatly indebted to L. Costa and J. Pons-Llopis for a long time and enjoyable collaboration which led to part of the material described in these notes.

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Authors and Affiliations

  1. 1.Department de Matemàtiques i InformàticaUniversitat de BarcelonaBarcelonaSpain

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