When is a homogeneous ideal a limit of saturated ones?

Abstract: Let I be a homogeneous ideal in a polynomial ring S. If the Hilbert function of S/I is admissible, for example (1,n,n,n,...) is it natural to ask whether I is a limit of homogeneous ideals: does there exist a ideal F in S[t] such that F(t = 0) is equal to I, while F(t = lambda) is a saturated homogeneous ideal for lambda general. Examples of such limits (for the above Hilbert function) can be constructed e.g. by degenerating I(Gamma), where Gamma is a tuple of n general points on the projective space associated to S. However, to decide whether a given ideal I is a limit is very much nontrivial. This problem very recently became of key interest for applications in the theory of tensors: proving that certain ideals are not limits would improve best known lower bounds on border ranks of certain important tensors. In the talk I will report how surprisingly little is known and present some recent results and some challenges, both theoretical and computational. All this is a joint work with Tomasz Mandziuk.

Mathematics

Audience: advanced learners

IIT Bombay Virtual Commutative Algebra Seminar

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