Valery Alexeev

Abstract:   Does a one-parameter family of (say, algebraic) varieties have a canonical limit? The answer depends on the category in which they are considered.

For embedded projective varieties there are three different approaches: varieties as cycles (van der Waerden), subschemes (Grothendieck), and varieties with a finite map (joint with Knutson), which lead to three different moduli spaces: Chow variety, Hilbert scheme, or moduli of branchvarieties.

For non-embedded varieties the Minimal Model Program gives the canonical limits, higher-dimensional generalizations of the Deligne-Mumford-Knudson-Kontsevich stable curves and maps. I will review several cases of special interest (Calabi-Yau varieties, etc.)

Tropical geometry gives a simplified picture of one-parameter degenerations. Using the above examples, I will explain what it captures and what it misses.

Here are some directions to Northeastern University. Lake Hall and Nightingale Hall can be best accessed from the entrance on the corner of Greenleaf Street and Leon Street. The two halls are connected, with no well-defined boundary in between. In particular, 509 Lake Hall is on the same corridor as 544 Nightingale Hall.

There is free parking available for people coming to the Colloquium at Northeastern's visitor parking (Rennaisance Garage). The entrance is from Columbus Avenue. If coming by car, you should park there and take the parking talon. After the lecture, you may pick up the payment coupon from Andrei Zelevinsky.