Period, Index, and an Invariant of Grothendieck for Relative Curves
نویسندگان
چکیده
Assume from now on that X/K is a smooth proper geometrically connected curve of genus g, and consider the degree morphism deg : P(K)→ Z. The period δ′(X/K) of X/K is defined to be the positive generator of the image of the degree map P(K)→ Z. The index δ(X/K) is the positive generator of the image of the degree map Pic(X) → Z. We let γ(X/S) denote the positive generator of the image of the composition P(S) → P(K) → Z. Clearly, it follows from the definitions that δ′(X) | γ(X ) and δ′(X) | δ(X). Assume in addition that X is regular. Then Pic(X )→ Pic(X) is surjective, and δ′(X) | γ(X ) | δ(X).
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