Sebastian Bozlee (Tufts)

Tuesday, March 8, 2022 - 1:30pm

PDL C-38

**Preseminar 1:30-2:30**

**Title**: An introduction to modular compactifications of moduli spaces of curves

**Abstract**:

In modern algebraic geometry, we often study algebraic varieties of some kind using a space that parametrizes all such objects: a moduli space. These moduli spaces tend not to be compact, which makes them more difficult to study. To remedy this, we look for ways to compactify them.

In this talk, we will explore these ideas by example. We will introduce the idea of a moduli space, paying special attention to the moduli space of curves of degree d in projective space. Then we will explore several extant compactifications of the moduli space of abstract pointed curves, paying special attention to genus zero curves with weighted points.

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

**Title**: A classification of Gorenstein compactifications of M_{1,n}

**Abstract**:

The moduli space of smooth pointed elliptic curves M_{1,n} is not compact. Earlier work by David Smyth has shown that for each n, M_{1,n} admits n distinct compactifications which are themselves moduli spaces of Gorenstein curves. This leads to the natural question: are these all such compactifications?

In this talk, we will present a new family of compactifications of M_{1,n} by Gorenstein curves coming from joint work with Adrian Neff and Bob Kuo. These new moduli spaces provably exhaust such Gorenstein modular compactifications of M_{1,n}, the first general classification of modular compactifications since 2012. Time permitting, we will explore the tropical and log geometric reasoning that led to the discovery of these spaces and our classification result.

In this talk, we will present a new family of compactifications of M_{1,n} by Gorenstein curves coming from joint work with Adrian Neff and Bob Kuo. These new moduli spaces provably exhaust such Gorenstein modular compactifications of M_{1,n}, the first general classification of modular compactifications since 2012. Time permitting, we will explore the tropical and log geometric reasoning that led to the discovery of these spaces and our classification result.

The talk will also be available via Zoom: https://washington.zoom.us/j/689897930

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