UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

January 21, 2020

939 Evans Hall

3:45PM: Enumerating pencils with moving ramification on curves

Carl Lian

We consider the general problem of enumerating branched covers of the projective line from a fixed general curve subject to ramification conditions at possibly moving points. Our main computations are in genus 1; the theory of limit linear series allows one to reduce to this case. We first obtain a simple formula for a weighted count of pencils on a fixed elliptic curve E, where base-points are allowed. We then deduce, using an inclusion-exclusion procedure, formulas for the numbers of maps E->P^1 with moving ramification conditions. A striking consequence is the invariance of these counts under a certain involution. Our results generalize work of Harris, Logan, Osserman, and Farkas-Moschetti-Naranjo-Pirola.

5:00PM: Singularities of generic projection hypersurfaces

Takumi Murayama

Classically, it is known that every algebraic variety over an algebraically closed field is birational to a hypersurface in some projective space. Using generic linear projections, one can show that this hypersurface can be taken to have at worst nodal singularities in dimension one, or at worst ordinary singularities in dimension two. We present generalizations of these results to arbitrary dimension. In particular, in dimensions up to five, we show that the resulting singularities are those that appear on degenerations of smooth varieties in moduli theory. This result is due to Doherty over the complex numbers, and is joint with Rankeya Datta in positive characteristic.

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