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WEIGHTED PROJECTIVE VARIETIES by Igor Dolgachev Contents 0. Introduction i. Weighted projective space i.i. Notations 1.2. Interpretations 1.3. The first properties 1.4. Cohomology of 0F(n) 1.5. Pathologies 2. Bott's theorem 2.1. The sheaves ~(n) 2.2. Justifications 2.3. ~$(n) 3. Weighted complete intersections 3.1. Quasicones 3.2. Complete intersections 3.3. The dualizing sheaf 3.4. The Poincare series 3.5. Examples 4. The Hodge structure on cohomology of weighted hypersurfaces. 4.1. A resolution of ~i X 4.2. The Griffiths theorem 4.3. Explicit calculation 4.4. Examples and supplements O. Introduction. In this paper I discuss the technique of weighted homogeneous coordinates which has appeared in works of various geometers a few years ago and it seems has been appreciated and armed by many people. In many cases this technique allows one to present a nonsingular algebraic variety as a hypersurface in a certain space (a weighted projective space) and deal with it as it would be a nonsingular hypersur- face in the projective space. A generalization of this approach is the technique of polyhedral projective spaces for which we refer to [5, 6, 15].
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85 Section 1 deals with weighted projective spaces, the spectrums of graded poly- nomial rings. Most of the results from this section Can be found in [7]. In section 2 we generalize the Bott theorem on the cohomology of twisted sheaves of differentials to the case of weighted projective spaces. Another proof of the same result can be found in [23] and a similar result for torical spaces is discussed in [5]. In section 3 we introduce the notion of a quasismooth subvariety of a weighted projective space. For this we define the affine quasicone over a subvariety and require that this quasicone is smooth outside its vertex. We show that quasismooth weighted complete intersections have many properties of ordinary smooth complete intersections in a projective space. The work of Mori [19] contains a similar re- sult but under more restrictive conditions. Rather surprisingly not everything goes the same as for smooth complete intersections. For example, recent examples of Catanese and Todorov show that the local Torelli theorem fails for some quasi- smooth weighted complete intersections (see [4, 24]). In section 4 we generalize to the weighted case the results concerning the Hodge structure of a smooth projective hypersurfaces. Our proof is an algebraic version of one of Steenbrink [23] and can be applied to the calculation of the De Rham cohomology of any such hypersurface over a field of characteristic zero. The present paper is partially based on my talks at a seminar on the Hodge-Deligne theory at Moscow State University in 1975/76. It is a pleasure to thank all of its participants for their attention and criticism. i. Weishted projective space. i.i. Notations Q = {qo,ql,. ..,qr } , - a finite set of positive integers; IQI = qo +-.
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This note was uploaded on 02/24/2012 for the course MATH 296 taught by Professor Igordolgachev during the Fall '06 term at University of Michigan-Dearborn.

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