Date: Thursday, April 04, 2013
Location: 3096 East Hall (3:00 PM to 4:00 PM)
Title: Ideals and algebras generated by forms of degree at most 4 in polynomial rings
Abstract: Michael Stillman raised the following question: given n homogeneous polynomials of degree at most d in a polynomial ring over a field, is there a bound on the projective dimension for the ideal they generate that depends on n,d and not on the number of variables? Hilbert proved that the number of variables is a bound. The talk will discuss joint work with Tigran Ananyan which answers this question affirmatively if the forms have degree at most 4, provided the characteristic of the field is not 2 or 3. Results of the following kind over an algebraically closed field play a key role: Given a vector space of quadratic forms of dimension 3, if no nonzero element is in the ideal generated by 27 linear forms, then the quotient by the ideal generated by the quadratic forms is a UFD. We can prove similar results up to degree 4 if the characteristic is not 2 or 3. I.e., for d at most 4, there are functions A(n,d) such that if o nonzero homogeneous element in the vector space spanned by at most n forms is in an ideal generated by A(n,d) elements of lower degree, then the quotient by the ideal generated by any subset of the forms is a unique factorization domain. We conjecture that such results hold in general, if the characteristic is 0 or greater than d.
Speaker: Mel Hochster
Institution: University of Michigan
Event Organizer:
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