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Let Sg, s > 0, g ≥ 2, be the moduli scheme of all (C,L1, . . . , Ls), where C is a smooth curve of genus g and each Li is a theta-characteristic. For all ri > 0, 1 ≤ i ≤ s, set [s]S1s g := {(C,L1, . . . , Ls) ∈ Sg : h (Li) = ri + 1 for all i}. Here we raise several questions on these algebraic sets. We show that if T is a general element of a component of [s]S1s g with the expect codimension ∑s i=1 ( ri+1 2 ) and (C,L1, . . . , Ls) is general in T , then each Li has no base points. AMS Subject Classification: 14H10, 14H51

CURVES WITH SEVERAL THETA-CHARACTERISTICS WITH A PRESCRIBED NUMBER OF SECTIONS: A QUESTION LIST