The Geometry of Schemes (Graduate Texts in Mathematics)

The Geometry of Schemes (Graduate Texts in Mathematics)

David Eisenbud

Language: English

Pages: 300

ISBN: 0387986375

Format: PDF / Kindle (mobi) / ePub


Grothendieck’s beautiful theory of schemes permeates modern algebraic geometry and underlies its applications to number theory, physics, and applied mathematics. This simple account of that theory emphasizes and explains the universal geometric concepts behind the definitions. In the book, concepts are illustrated with fundamental examples, and explicit calculations show how the constructions of scheme theory are carried out in practice.

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at infinity on the y-axis) as the zero locus of an equation y − ax − b; a and b are then affine coordinates on the corresponding subset A2 ⊂ K (P2 )∗. The equations of the universal line section Γ K C of C in the open subset A2 × A2 ⊂ P2 × (P2 )∗ are then K K K K ΓC = V (y − ax − b, y2 − t(x2 − 1)) ⊂ Spec K[a, b, x, y] = V ((ax + b)2 − t(x2 − 1)) ⊂ Spec K[a, b, x]. Now, we may expand out the equation of ΓC as (ax + b)2 − t(x2 − 1) = (a2 − t)x2 + 2abx + (b2 + t), from which we see

proves the existence of a solution that is a distribution, and then is left with the (possibly more tractable) regularity problem of proving that the distribution is represented by integration against a function. (3) Many aspects of the geometry of schemes can be extended to the cat- egory of functors, so that it is sometimes useful to forget about repre- senting a functor and work in that category (or some suitable subcat- egory) directly. In this chapter we illustrate these points,

apart from giving a more natural definition, this characterization of the Fano schemes will allow us to determine their tangent spaces. Now suppose we are given two projective S-schemes X ⊂ Pm S and Y ⊂ PnS. We may embed the product X ×S Y in projective space via the Segre map X ×S Y → Pm × → PN, S S Pn S S where N = (m + 1)(n + 1) − 1. We thus have Hilbert schemes parametrizing subschemes of a product X ×S Y . This in turn allows us to parametrize morphisms from X to Y , by con-

(I) ⊂ X into an affine scheme by identifying it with Y = Spec R/I. This makes sense because the primes of R/I are exactly the primes of R that contain I taken modulo I, and thus the topological space |Spec R/I| is canonically homeomorphic to the closed set V (I) ⊂ X. We define a closed subscheme of X to be a scheme Y that is the spectrum of a quotient ring of R (so that the closed subschemes of X by definition correspond one to one with the ideals in the ring R). We can define in these

able to write down the Hilbert polynomial of a complete intersection in a much more transparent way. The Hilbert series HM (t) of a module M is easy to define: if P (M, ν) is the Hilbert function of M, we let HM (t) be the Laurent series ∞ HM (t) = P (M, ν)tν . ν=−∞ We define the Hilbert series HX (t) of a subscheme X ⊂ Pn to be the K Hilbert series of its coordinate ring SX = S/I(X). The first thing to note is that the Hilbert series of projective space itself is simple: we have 1

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