By Alexander Degtyarev, Ilia Itenberg, Viatcheslav Kharlamov (auth.)

This is the 1st try of a scientific research of actual Enriques surfaces culminating of their class as much as deformation. uncomplicated specific topological invariants are elaborated for opting for the deformation periods of actual Enriques surfaces. a few of theses are new and will be utilized to different periods of surfaces or higher-dimensional kinds. meant for researchers and graduate scholars in actual algebraic geometry it might additionally curiosity others who are looking to familiarize yourself with the sector and its recommendations. The examine is determined by topology of involutions, arithmetics of necessary quadratic kinds, algebraic geometry of surfaces, and the hyperkähler constitution of K3-surfaces. A complete precis of the mandatory effects and methods from every one of those fields is incorporated. a few effects are constructed extra, e.g., an in depth research of lattices with a couple of commuting involutions and a definite category of rational complicated surfaces.

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R. 7) we may assume that m. M =0 l. xl"",x d for some i=l. Proof ov induction on (O:x l ) = M and hence M e R(x 1 ,···, xd IM) = e R/ Xl (xZ,···,xdIM) - e R/ Xl (xZ,···,xdIM) Now suppose that d = s. We have by d =s + (J.. 5) , = o. 1, s > 0 and the result holds for 28 Since xlm-l( xlM ) = xl(M/xlM) = 0, by induction the result follows. 9) Let M be any finitely generated R-module and be a system of parameters for PROOF. xl, ••. ,x d R. Then First, by induction on d, we show eR(x l , ••• ,xdIM) 2: O.

U«xl)+U(O» for n. Put and '1€ Ass(R/«x1)+U(0») «Xl)+(O:X~»' = «xl)+U(O»~ Moreover, larger <==> n 1Rk R' := R/(Xl)+(O:X~) = (x k ) + U(1n k_l ) induction we get for large n, ~i = (0) for any 1 < k eO(q;R) = eO(q';R') = 111 = U(O) + (xl)' it follows that ~ d. Then by i(R'/~d). Now, since 42 U( m1) =U( U(O)+(x l ) ) =U( (xl )+(0: X~» and 'Uld= ~/«xl )+(0: x~») for large n. Therefore eO(qiR)=eO(q'iR')=t(R'/1nd}=t(R/1nd). 26) EXAMPLE. 126]). v 1 'f-v 2 = We put A(Vl;C) := A ~V·A = (X l ,X 4 )A is generated by a system of parameters Then 2 Xl ,X 4 for + (Xl) = (Xl'X 2X3 , X2 , X2X4 - X3 ) n (Xl'X 2 ,X 3 ,X 4 ) 2 fI[) ~V A and 2 2 is a primary decomposition of ~V 2 NJ 2 3 1 3 3 + (Xl) in A.

PROOF. (i) Clear. - K-dim (A/I). It is enough to prove that, for every minimal prime ideal q of (I,x) K-dim (A/q) = d-l. 56 Since I is unmixed d = K-dim (A/I) = K-dim every 1 e: ideal of of I Ass (A/I). Let (A/~) for (I,x) c q CA be a minimal prime (I,x). Then there exists a minimal prime ideal such that ~ ~ q and by Krull's Principal Ideal ht q/1 = 1. we get K-dim A/q = K-dim (iii) Let such that K-dim since I - ht q/3? = d - 1. A/~ I C q C R be a minimal prime ideal of = K-dim (R/q)~ is unmixed and R/q (R/I)~ I = K-dim (A).