By Barletta E., Dragomir S., Duggal K.L.

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**Example text**

8], also [17]). For higher values of the genus i) M is known to be unirational g uniruled for g = I! 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.

Clearly given by F(t), where _S4 on t 2 = (w21) (w2)(w3). 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.

Before auxiliary turning RATIONALITY to prove OF THE INVARIANT the rationality of SUBFIELDS. R4, we first state a more general result. Let V be the standard permutation representation of the symmetric group S n -n' the direct sum of m copies of V . Xln , . x21 , . X2n, . . Xmn ~, and a perm Vn mutation ~ acts on ant rational xij functions, and a variable by sending it to xi~(j ). where o. 2. m. 1 form a basis is a rational of the purely field: more precisely transcendental extension k. o', 0 (2) .