On large theta-characteristics with prescribed vanishing

Let C be a smooth projective curve of genus g≥ 2. Fix an integer r≥ 0 , and let k̲=(k_1,…,k_n) be a sequence of positive integers with ∑k_i=g-1. In this paper, we study n-pointed curves (C, p_1, … , p_n) such that the line bundle L:=OC(∑k_ip_i) is a theta-characteristic with h(C, L) ≥ r+ 1 and h0(C,L)≡r+1(mod2). We prove that they describe a sublocus G_g^r(k̲) of M_{g,n} having codimension at most g-1+r(r-1)/2. Moreover, for any r≥ 0 , k̲ as above, and g greater than an explicit integer g(r) depending on r, we present irreducible components of G_g^r(k̲) attaining the maximal codimension in M_{g,n}, so that the bound turns out to be sharp.