By Donu Arapura
This can be a rather fast-paced graduate point advent to advanced algebraic geometry, from the fundamentals to the frontier of the topic. It covers sheaf concept, cohomology, a few Hodge conception, in addition to a number of the extra algebraic points of algebraic geometry. the writer often refers the reader if the therapy of a undeniable subject is quickly on hand in different places yet is going into substantial element on issues for which his therapy places a twist or a extra obvious standpoint. His circumstances of exploration and are selected very conscientiously and intentionally. The textbook achieves its goal of taking new scholars of complicated algebraic geometry via this a deep but large advent to an unlimited topic, ultimately bringing them to the vanguard of the subject through a non-intimidating type.
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Extra resources for Algebraic Geometry over the Complex Numbers (Universitext)
The germ at x of a function f deﬁned near X is the equivalence class containing f . We denote this by fx . 1. Given a presheaf of functions P, its stalk Px at x is the set of germs of functions contained in some P(U) with x ∈ U. It will be useful to give a more abstract characterization of the stalk using direct limits (which are also called inductive limits, or ﬁltered colimits). We explain direct limits in the present context, and refer to [33, Appendix 6] or  for a more complete discussion.
Suppose that Γ is a group of diffeomorphisms of a manifold X. Suppose that the action of Γ is ﬁxed-point-free and properly discontinuous in the sense that every point possesses a neighborhood N such that γ (N) ∩ N = 0/ unless γ = id. Give Y = X/Γ the quotient topology and let π : X → Y denote the projection. Deﬁne f ∈ CY∞ (U) if and only if the pullback f ◦ π is C∞ in the usual sense. Show that (Y,CY∞ ) is a C∞ manifold. Deduce that the torus T = Rn /Zn is a manifold, and in fact a Lie group. 20.
14. A point x on a (not necessarily afﬁne) variety X is called a nonsingular or smooth point if dim TX,x = dim X; otherwise, x is called singular; X is nonsingular or smooth if every point is nonsingular. The condition for nonsingularity of x is usually formulated as saying that the local ring OX,x is a regular local ring, which means that dim OX,x = dim Tx [8, 33]. But this is equivalent to what was given above, since dimX coincides with the Krull dimension [8, 33] of the ring OX,x . Afﬁne and projective spaces are examples of nonsingular varieties.