A Conjecture of Mori and Families of Plane Curves

Consider a smooth family of hypersurfaces of degree d in P^{n+1}.
When is every smooth projective limit of this family also a hypersurface?
While it is easy to construct example of limits that are not
hypersurfaces when the degree d is composite, Mori conjectured that, if d
is prime and n>2, every smooth projective limit is indeed a hypersurface.
However, there are counterexamples when n=1 or 2; for example, one can
take a family of degree 5 plane curves and degenerate to a smooth
hyperelliptic (non-planar) curve.  In this talk, we will propose a
re-formulation of Mori’s conjecture that explains the failure in low
dimensions, provide results in dimension one, and discuss a general
approach to the problem using moduli spaces of pairs.  This is joint work
with David Stapleton.