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8320homework3 - HOMEWORK PROBLEMS FOR MATH 8320 48-71 PETE...

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HOMEWORK PROBLEMS FOR MATH 8320: 48-71 PETE L. CLARK This is the third set of homework problems for the 8320 course. As usual, unless mention is made to the contrary, k is assumed to be an arbitrary field. However, we are content to develop the theory of hyperelliptic curves only in characteristic different from 2, and there is only one problem where the imperfection of the ground field is addressed in an important way. Note: Perhaps unfortunately, this problem set contains few problems of medium difficulty. Rather, most of the problems are either easy, hard or (so far as I know), unsolved. In order to make your work more efficient, I have labelled parts of prob- lems with a designation of (E), (M), (H) or (U), accordingly. Do understand that this classification is highly surjective. Exercise 48(E): Show that the index of a variety is equal to the greatest com- mon divisor of all degrees of closed points P on V . (Recall that our definition of the index was as the cardinality of the cokernel of the degree map from Z 0 ( V ) to Z .) Exercise 49: (a(E)) If k is not perfect, we can also define the separable index i s to be the gcd of all degrees of closed points with separable residue field extensions. What is an equivalent definition in terms of divisors? (Hint: you want to define a subgroup of Z 0 ( X ) generated by “separable” divisors.) (b(E)) Show that i ( V ) | i s ( V ) for all V . (c(U)) Find a nice variety V over some (imperfect) field k such that i S ( V ) > i ( V ). Exercise 50(M): Show that the Weierstrass points on a hyperelliptic curve of genus g 2 are precisely the ramification points of the covering C C/ι = P 1 . In particular, there are exactly 2 g + 2 Weierstrass points. Exercise 51(H): In this exercise you will fill in the details of a statement and proof of Hurwitz’s L¨uckensatz . Exercise 52: Let k be an imperfect field and C /k a nice (in particular, geomet- rically regular) curve. (a(E)) Must every automorphism of C be defined over a separable extension of k ? (b(H)) What about if we assume that the genus is at least 2?
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