## University of Sydney Algebra Seminar

# David Eisenbud (University of California, Berkeley)

## Monday 13th July, 12.05-12.55pm, Carslaw 375

### Plato's Cave: What we still don't know about linear projections

The theorem that any finite separable field extension has a primitive element (which can be taken to be a linear combination of a given set of generators) has a geometric version: Any algebraic variety (in characteristic zero) can be mapped onto a hypersurface by a linear projection that is an isomorphism on some open dense subset.

How much is the variety necessarily changed by such a projection? This classical question is well understood in low dimension, and very little understood in high dimension.

I'll explain how the question arises in classical and modern algebraic geometry. Then I'll describe some of what is known, and what one might hope. The new parts of what I describe are from recent work with Roya Beheshti.