Math 254B. Homework #7

Due Tuesday 13 April, online

1. Show that the special case k = ℚ, S = {∞} of the general Roth’s theorem (as presented in class on Monday, 5 April) is equivalent to the version of Roth’s theorem presented earlier in the semester.

2. Let k be a number field, let E be the curve y^{2}z = x^{3} − xz^{2} in ℙ_{k}^{2} , and let D be the divisor consisting of the point with homogeneous coordinates [0 : 0 : 1] (with multiplicity 1 ). (These are the same curve and divisor as in Question 2 on Homework 6.)

Use “min-max” to find an explicit Weil function for D.

3. Let k and Y be as in class, and let V be a variety over k (not necessarily quasi-

projective).

(a). Show that, for any line sheaf ℒ on V , there exists a model X for V over

Y such that ℒ extends to a line sheaf on X. (b). Do the same for a Cartier divisor D on V .

[This partially completes the proof of Lemma 2.8 in the handout “Chapter 3: Models.”]