## Algebraic spaces |

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affine scheme affine space algebraic family algebraic functions algebraic structure algebraic surface analytic space arbitrary birationally equivalent blowing-up closed subscheme coefficients complete algebraic spaces complete local ring complex analytic space condition Consider contracted convergent series coordinate functions definition diagram dimension effective formal moduli equivalence relation examples exceptional curve exist F has effective F k[t finite local k-algebra finite presentation finite set finite type finite-to-one formal algebraic space formal completion formal modification formal power series functor functor F global functions Hilbert homomorphism infinitesimal neighborhoods intersection numbers isomorphism let f locally of finite map f map of affine maximal ideal modification f module Moisezon space normal bundle notion obtained open subspaces polynomial equations problem projection maps projective scheme quotient reduced scheme resolution of singularities ring of algebraic ring of global series solution y(x Spec surjective map system of polynomial Theorem 1.7 underlying point set unique universal deformations vector space zero