Commutative Algebra for Singular Algebraic Varieties
O. E. Villmayor U.

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The aim of this course is to discuss some of the algebraic background concerned with resolution of singularities; and the task is to somehow balance the formalism, required to present ideals, with the applications to concrete results.

Some ideas deserve a thorough discussion, but then there are also results which are convenient to simply formulate and to see their use.

The course begins with the discussion of the blow up of ideals, a subject with implications and interest in various elds.

It is a usual pattern in algebraic geometry to produce objects, or say global data, which are dened by patching local data. The blow up of an ideal is only one example of this, in fact it is a particular case of a projective scheme. We shall address this issue by discussing the patching of local data. At some point some properties of schemes will be formulated, just enough to carry on the discussion.

Within the first week we will study projective spaces, projective morphisms, blow-ups of ideals, and overview their universal properties.

In the second week attention will be drawn to analytic methods. Here we consider algebras over a perfect eld, and discuss properties derived from the use of dierential operators on smooth algebras. This parallels the role of partial derivatives in the study of singularities of a hypersurface dened by a polynomial over a eld. Polynomial rings are, in fact, the main example of smooth algebras.

Some applications of the previous subjects will be discussed in the third week. The notion of multiplicity, and its expression in terms of generic projections, is an interplay between algebra and geometry. We shall also discuss, within this frame, the denition of invariants for the resolution of singularities.

Syllabus

• (1) Projective morphisms, blow-up of ideals.
• (2) Varieties, smooth schemes over perfect elds.
• (3) Regular rings and monoidal transformations.
• (4) Higher order dierentials, with applications to singularities.
• (5) Rees algebras of ideals on smooth schemes. Integral closure and differential operators.
• (6) Projections, multiplicity theory, and elimination of variables.
• (7) Local Global aspects in the denition of invariants for algorithmic resolution.

Some introductory notes are to be posted later.

Reading list

• S. D. Cutkosky, Resolution of singularities, Graduate Studies in Mathematics, 63, Providence, R.I. : American Mathematical Society, 2004.
• R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer- Verlag 1977.
• M. Reid, Undergraduate Commutative Algebra, London Mathematical Society Student Texts 29, Cambridge University Press 1995.
• I. S. Shafarevich, Basic Algebraic Geometry 1, 2, second edition, Springer- Verlag 1994.
• K.E. Smith, L. Kahana, P. Keaaainen, W. Traves, An Invitation to Algebraic Geometry, Universitext, Springer-Verlag, 2000.

Prerequisites for the course

The students should master, or at least should have some acquaintance, with the contents in chapters 4,5, and 6, of Reid's Undergraduate Commutative Algebra.