APS
Speaker: Amanpreet Singh, IIT Bombay
Host: Saurav Bhaumik
Title: Moduli Field Vs. Field of Definition of projective schemes over $\mathb{C}$
Time, day and date: 4:00:00 PM – 5:00:00 PM, Tuesday, September 29
Venue: Ramanujan Hall
Abstract: In this seminar, we study the moduli field $M(X)$ of projective schemes $X$ and the question of when a projective scheme can be defined over a finite extension of its moduli field.
we study this question for projective schemes whose canonical sheaf is ample or anti-ample, and prove the following main theorem: Let $X$ be a smooth, irreducible projective scheme of finite type over $\mathbb{C}$ and the canonical sheaf $\omega_X$ or the anti-canonical sheaf $\omega_X^{-1}$ is ample, then X can be defined over a finite extension of its moduli field M(X).
As an immediate consequence, we recover the theorem of Hammer and Herrlich: a smooth projective curve over $\mathbb{C}$ of genus $g \geq 2$ can be defined over a finite extension of its moduli field.