By I.R. Shafarevich (editor), R. Treger, V.I. Danilov, V.A. Iskovskikh

This EMS quantity comprises components. the 1st half is dedicated to the exposition of the cohomology conception of algebraic kinds. the second one half bargains with algebraic surfaces. The authors have taken pains to offer the cloth conscientiously and coherently. The ebook comprises quite a few examples and insights on a number of topics.This ebook may be immensely beneficial to mathematicians and graduate scholars operating in algebraic geometry, mathematics algebraic geometry, advanced research and comparable fields.The authors are famous specialists within the box and I.R. Shafarevich is usually recognized for being the writer of quantity eleven of the Encyclopaedia.

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**Example text**

J S(Zl) = z][, s(z 2) = -z 2. Step III. k(V) r is a purely transcendental extension of k, K, generated by 4 2 2 2 2 Xl' Xl/x 2' Xl/x 3' XlYl' Yl/y 2' Yl/ ' Zl(Xl+Y;)' z2(xl+Yl )" Y3 Proof. " by s s(u) = I/u: End of the proof. finally we replace x? E K, I/Y 1 4 to avoid this u by I/xlY 1 (u-l)/u+] = w In this way we have a linear action of al vector space, and with so we have equality. u = x2/ 2 = x4/ 2 2 . 1 Then but is not linear any more: eigenvalues equal to The quotient is obviously rational.

For e 40 ([9]), g, for the situation is as follows: g ~ |0, whereas, for ([|6], [I]), g o d d ~ 25 g = 12 M ([|4J), is variety of gen- g eral type ([7]), and D. Mumford and J. Harris announced a similar result also ii) g even the unirationality of R g for g = 5,6 has been proven only recently ([4], [6]). If the base field is of characteristic degree *) 22g - I, so that theorems A and ~2, C R is a covering of M of g g produce two rational coverings of Part of this research was done when the author was at the Institute for Advanced Study, partially supported by NSF grant MCS 81-033 65.

1. REFERENCES [1] P. Deligne and D. Mumford, The irreducibility genus. Publ. IHES 36 (1969), 75-109. [el J. Harris, A bound on the geometric genus of projective varieties. Norm. Sup. Pis Serie IV, vo. VII, 1 (1981), 35-68. of the space of curve of given Ann. Sci. 1] sub- sDace. submani~olds, Z, N ( M ) , a trivial of of If T*Z. X let C c M Put should space 0X c T * Z l x Now, by a paper the G r a s s m a n n of A. e. 8], fN 3, b e l o w . set of bundle of talk was motivated who worked N*(X0). complex group, of a m p l e n e s s .