Kenji Ueno's Algebraic Geometry 2: Sheaves and Cohomology PDF

By Kenji Ueno

Sleek algebraic geometry is outfitted upon primary notions: schemes and sheaves. the speculation of schemes was once defined in Algebraic Geometry 1: From Algebraic kinds to Schemes, (see quantity 185 within the related sequence, Translations of Mathematical Monographs). within the current booklet, Ueno turns to the idea of sheaves and their cohomology. Loosely conversing, a sheaf is a fashion of keeping an eye on neighborhood info outlined on a topological area, corresponding to the neighborhood holomorphic features on a fancy manifold or the neighborhood sections of a vector package. to review schemes, it truly is valuable to review the sheaves outlined on them, specifically the coherent and quasicoherent sheaves. the first instrument in knowing sheaves is cohomology. for instance, in learning ampleness, it truly is usually precious to translate a estate of sheaves right into a assertion approximately its cohomology.

The textual content covers the $64000 themes of sheaf thought, together with kinds of sheaves and the basic operations on them, reminiscent of ...

coherent and quasicoherent sheaves.
proper and projective morphisms.
direct and inverse pictures.
Cech cohomology.

For the mathematician strange with the language of schemes and sheaves, algebraic geometry can look far-off. in spite of the fact that, Ueno makes the subject look typical via his concise variety and his insightful reasons. He explains why issues are performed this fashion and supplementations his motives with illuminating examples. for that reason, he's in a position to make algebraic geometry very available to a large viewers of non-specialists.

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Extra info for Algebraic Geometry 2: Sheaves and Cohomology

Sample text

The first contains a line, determinable in K, from which it evidently projects into a rational quintic scroll, birational in K, while the second*) projects from a tangent plane into a surface of VERONESE type, since its plane representation is by quartics

Also st passes through the base group (Q(1 5), which does not lie on st*. Through any point P distinct from (Q(IS) and (Q(j 5) there passes a single curve of each of the pencils if>, if>1> if>~. If P is common to st, st*, it is double for the curve Q( of if> which contains it, i. e. it lies on Bs; or it is a contact of this curve with each of the curves of if>1> if>j passing through P; and conversely. In this last case these two curves can coincide with ~, and then P is double for the g(~~l cut by if> on ~; otherwise P lies on 9\; and conversely.

By virtue of these theorems we may speak of a rational curve or surface whenever we are not concerned with the particular extension K' of K in which its birational representation is obtainable. We shall see later that there exist irrational (i. e. non-birational) involutions in any space Sd (d;;::; 3), so that a variety Va (d;;::; 3) which is unirational in K' is not in general birational in any extension oj K'. We illustrate these ideas by examples which will prove useful in the sequel. To begin with, any monoid (primal of order n with a single (n -I)-pIe point) is obviously birational in K; hence, in particular, the VERONESE surface is birational in K; for its general projection on 53 is a monoid (STEINER surface).

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Algebraic Geometry 2: Sheaves and Cohomology by Kenji Ueno


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