Download PDF by John von Neumann: Functional operators: geometry of orthogonal spaces

By John von Neumann

Measures and integrals

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HOz generated by the elements ofthe form 0:01-100:. z ~ Oz. z/l'iz' which we view as an Oz-module (or a coherent sheaf on Z) via the first projection. z / l'i z ' we define a . (3 to be (pri a) (3 = (a 0 1)(3. e. an irreducible and reduced k-scheme. e. the completion Oz of Oz (with respect to the maximal ideal iJJ1 z ) is isomorphic to the formal power series ring k[[Zl' ... ,znll at every closed point Z E Z, where n = dim Z. In fact, we have checked that 01,z is of rank n at a smooth point z. Conversely, if 01 , z is of rank n, then iJJ1 z /iJJ1;, and hence 0 z = lim 0 z /iJJ1~, is generated by n +-- 01, 01 01 01 01 elements.

4 in case B =F 0. (2) Let Y be a closed subscheme of a smooth projective variety X. Assume that Y is locally a complete intersection so that I y / I~ is a locally free sheaf on Y. Use a similar argument as above and prove that the Zariski tangent space of Hilb(X) Lecture II Construction of Non-Trivial Deformations via Frobenius 31 at [Y] is naturally identified with Homo x (Iy / I~, Oy) ~ HO (Y, Ny / x), where Ny / x = Homo y (Iy / I~, Oy) is the normal bundle. Hint: Take a closed subs cherne Y C Speck[E]j(E2) x X, flat over the base Speck[E]/(E2).

Our proof depends completely on the intersection theory on a blown-up ruled surface W. Let us fix the notation. Let B = {Yl, ... , Yd} be the base point set. The morphism W --* 5 x C is a composite of blowing-ups /La with exceptional divisors Ea over smooth points Pa. Since two blowing ups commute each other if the centres 26 Part I: Geometry of Rational Curves on Varieties are mutually disjoint, we may assume that W is the fibre product of morphisms vI,ยท .. ,Vd and Va, where Vi is a composite of blowing ups with centers over Yi E C (i = 1, ...

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Functional operators: geometry of orthogonal spaces by John von Neumann

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