What is the Hilbert function of a projective scheme X of linear configurations? This is the question we will mainly deal with in this talk. In order to determine the Hilbert function of X in degree d, there is an expected value for it given by a naive count of conditions, which geometrically means that X imposes independent conditions on the linear system of hypersurfaces of degree d containing X. So the question becomes: when does X have the expected Hilbert function, and if it does not, why not? Interestingly enough, there is yet no general answer to it, in fact the problem remains widely open, even in dimX = 2 for re- duced schemes, and even in dimX = 1 for non-reduced schemes. We will discuss the classical and recent developments on the subject and the techniques involved, including our results regarding the non-reduced schemes.

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Meeting ID: 908 611 6889
Passcode: 362880