# Student Algebraic Geometry: Families of Elliptic Curves and the Canonical Bundle Formula

Hyunsuk Kim

Ruled surface is one of the most concrete examples we see when studying algebraic surfaces. These are surfaces that admit a fibration by $\PP^{1}$ over a curve. However, ruled surfaces are very boring since they are too easy in various ways. For instance, they can be easily classified, there are no degenerations, the algebraic structure of the fibre does not change, and the canonical bundle is easy to describe.

On the other hand, fibration by elliptic curves is way more entertaining since there are lots of things happening! We can study how the algebraic structure varies, how the fibre degenerates to a singular one, and can describe the canonical bundle in terms of the singular fibres and the moduli.

It turns out that these phenomena for elliptic surfaces can be generalized to many deep results in algebraic geometry such as variation of Hodge structure, degeneration of Hodge structure, adjunction and subadjunction, canonical bundle formula, semipositivity theorems, volume asymptotics and so on.

Despite the fact that elliptic fibrations are related to these profound theories in algebraic geometry, the example itself is very classical and can be understood explicitly. I will talk about these phenomena for elliptic surfaces in various perspectives.

On the other hand, fibration by elliptic curves is way more entertaining since there are lots of things happening! We can study how the algebraic structure varies, how the fibre degenerates to a singular one, and can describe the canonical bundle in terms of the singular fibres and the moduli.

It turns out that these phenomena for elliptic surfaces can be generalized to many deep results in algebraic geometry such as variation of Hodge structure, degeneration of Hodge structure, adjunction and subadjunction, canonical bundle formula, semipositivity theorems, volume asymptotics and so on.

Despite the fact that elliptic fibrations are related to these profound theories in algebraic geometry, the example itself is very classical and can be understood explicitly. I will talk about these phenomena for elliptic surfaces in various perspectives.

Building: | East Hall |
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Event Type: | Workshop / Seminar |

Tags: | Mathematics |

Source: | Happening @ Michigan from Student Algebraic Geometry Seminar - Department of Mathematics, Department of Mathematics |