# Download Algebraic Geometry: Proceedings of the Third Midwest by D. Burns (auth.), I. Dolgachev (eds.) PDF

By D. Burns (auth.), I. Dolgachev (eds.)

**Read or Download Algebraic Geometry: Proceedings of the Third Midwest Algebraic Geometry Conference held at the University of Michigan, Ann Arbor, USA, November 14–15, 1981 PDF**

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**Additional info for Algebraic Geometry: Proceedings of the Third Midwest Algebraic Geometry Conference held at the University of Michigan, Ann Arbor, USA, November 14–15, 1981**

**Sample text**

These covers are constructed with elementary arguments For theorem found in A) I use a classical in the case of theorems B),C) result of Wirtinger ([]8]), which can also be [5], and about which I was told by S. Recillas, (cf. 5. Our notation is as follows: k is an algebraically X is a complete Pic(X) closed field of char. #2 smooth curve of genus is the group of divisors on by g X defined over k modulo linear equivalence, here denoted m. e. 2n m0, n ~ 0. e. is the linear system of effective group in n letters, 0y(Ky) ~ ~y.

Theorem B. Proof. tion M4, 1 is a rational variety. 19 and the arguments preceding it we have a linear representa0:D 4 ÷ Aut(U) where U is l]-dimensional, nal to ~(U)/D4. 6) s(x,y) = (y,x), To decompose assume For s For sr, U r, s M4, ] isbiratio- 2 2 2 2 2 2 I, x, x , y, y , xy, x y, xy , x y , are the generators of D4, such that r(x,y) = (y,l-x). as a direct sum of irreducibles, char(k) # 2, and we know that we compute the character X since of D4 has order 8 and we O. we observe that O(s) permutes the elements of the basis, leaving 22 I, xy, X y fixed: hence X(S) = 3.

X4), extension of That M as _S4 on differs V3 k, and if T (yi) = yj, ~4 v wi' wiw i' Yi M ~ k(V~), (i=1,2,3). then M is a k(Sym2V4)~4 = M~3(t,(7), where t = w|w2w 3. k (Sym2V4)~4 = (M(t,(7))~3 , beginning, while 2 be the field generated by purely transcendental Proof. l • (w i) = +wj, hence T(wi) J) Step IV. t = WlW2W 3. ~3 = ~4/G" t # F, from the one of F t is an is isomorphic to ~3 but o is an invariant for follows by step III. nal field with basis of transcendency We conclude observing that from the very invariant by the formulas written in step I.