Eric Riedl

Biography

I am an algebraic geometer, interested in birational geometry, rational curves, hypersurfaces, and rationality problems. Specifically, I am interested in studying the geometry of hypersurfaces in projective space by studying the rational curves lying on them.

A motivating theme in algebraic geometry is that the positivity or negativity of the canonical bundle controls the geometry of the variety. There are many important, open conjectures in this vein like the Lang Conjectures and Manin's Conjecture, and an important test case for these conjectures are hypersurfaces in projective space. I study questions such as:

• As the degree of a very general hypersurface in Pⁿ changes, what happens to the dimensions of its spaces of rational curves? There are very precise conjectures about how they should behave, but there remain many important cases where this is unproven (such as Clemens' Conjecture for quintic threefolds).
• Do very general hypersurfaces in Pⁿ with degree d < n contain many rational surfaces? It is suspected that for n large and d near to n, the answer is no, which would provide an example of a rationally connected variety that is not unirational. This would resolve a longstanding open question in birational geometry. An important tool in understanding this question is to understand the normal bundles of these curves, which control the local structure of these moduli spaces.
• What do the entire curves on a very general large-degree hypersurface look like? As d gets to about 2_n, there are expected (by the Kobayashi Conjecture) to be none, making these hypersurfaces hyperbolic. However, the current bounds known are multiply exponential in n.