Scrollar invariants of smooth projective curves

Let X be a smooth projective curve of genus ggreater-or-equal, slanted3 and Rset membership, variantPick(X) with h0(X,R)=2 and R spanned. There are k−1 integers ei, 1less-than-or-equals, slantiless-than-or-equals, slantk−1, with e1greater-or-equal, slantedcdots, three dots, centeredgreater-or-equal, slantedek−1greater-or-equal, slanted0 and e1+cdots, three dots, centered+ek−1=g−k+1 associated to R (the so-called scrollar invariants of R). Here we prove that for all integers kgreater-or-equal, slanted3 and egreater-or-equal, slanted0 there is an integer g(k,e) such that for all integers ggreater-or-equal, slantedg(k,e) and all integers ei, 1less-than-or-equals, slantiless-than-or-equals, slantk−1, with e1greater-or-equal, slantedcdots, three dots, centeredgreater-or-equal, slantedek−1greater-or-equal, slanted0, e1less-than-or-equals, slantek−1+e and e1+…+ek−1=g−k+1 there exist a smooth curve X of genus g and Rset membership, variantPick(X) such that e1, …, ek−1 are the scrollar invariants of R.