Algebra and Geometry Seminar

The space of holomorphic maps from compact Riemann surfaces to a complex projective manifold X (in a fixed genus and degree) is typically non-compact, but a theorem of Gromov (refined by Kontsevich) shows that any sequence in this space always has some subsequential limit which is a "stable map" (i.e., a holomorphic map to X, defined on a compact complex curve with at worst nodal singularities, and having finite automorphism group). The compact moduli space of stable maps forms the foundation of the Gromov-Witten theory of X. Interestingly, it turns out that most stable maps which have "ghosts" (i.e., irreducible components of the domain on which the map is constant) can neverappear as the limit of any sequence described above. I will explain this background and then describe recent work (joint with Fatemeh Rezaee), where we produce a large class of stable maps (with ghosts of any genus) and give a constructive proof that these do indeed occur as limits. The proof relies on the study of explicit model cases and on general results from deformation theory.