EMBEDDING POINTED CURVES IN K3 SURFACES

Let D be a smooth projective curve of genus g over an algebraically closed field of characteristic zero. LetMg denote the moduli stack of such curves. Let (S, h) be a polarized K3 surface of genus g, i.e., h is ample and primitive with h2 = 2g − 2. Let Fg denote the moduli stack of such surfaces. Now suppose D ⊂ S with [D] = h