Kristin DeVleming (UMass Amherst)

Tuesday, October 25, 2022 - 2:00pm

PDL C-38

**Pre-seminar 2-2:30**

**Title**: Explicit degenerations of curves and surfaces

**Abstract**: If you've heard about moduli spaces of curves or other varieties, you've probably heard that we degenerate smooth things to singular things. How do we know what singular things appear? How do we actually write down degenerations of varieties? We'll go over several examples of degenerating families of curves that indicate a prototypical "wall crossing" behavior for moduli spaces.

**Seminar 2:30-3:20**

**Title**:A question of Mori and families of plane curves

**Abstract:**Consider a smooth family of hypersurfaces of degree d in P^{n+1}. An old question of Mori is: when is every smooth limit of this family also a hypersurface? While it is easy to construct examples where the answer is "no" when the degree d is composite, there are no known examples when d is prime and n>2! We will pose this as a conjecture (primality of degree is sufficient to ensure every smooth limit is a hypersurface, for n > 2). However, there are counterexamples when n=1 or 2. In this talk, we will propose a re-formulation of the 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.