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
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 , 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.