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Description: |
Almost thirty years ago, Herb Clemens made a series of conjectures about rational curves on general projective hypersurfaces, the most famous of which predicts that every rational curve on a general quintic threefold is rigid, in the sense of deformation theory. In this talk we will discuss a proof of the latter conjecture for rational curves of degree at most 11, which relies on the combinatorics of Borel-fixed ideals. We will also sketch a proof of the rigidity of rational curves on a suitably-chosen tropical quartic surface, which may be viewed as a first step towards proving the analogous statement for a tropical quintic threefold.
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