Exercise 111.46.6. Let k be an algebraically closed field. Let X be a smooth projective curve over k. Let r \geq s be integers. Show that there is a natural closed subset
such that a closed point (x_1, \ldots , x_ r, y_1, \ldots , y_ s) of X^ r \times X^ s is in Z if and only if x_1 + \ldots + x_ r - y_1 - \ldots - y_ s is linearly equivalent to an effective divisor. Hint: Choose an auxiliary invertible module \mathcal{L} of very high degree so that \mathcal{L}(-D) has a nonvanshing section for any effective divisor D of degree r. Then use the result of Exercise 111.46.5 twice.
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