Seminar

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Let P be a non-degenerate homogeneous prime ideal of height h in a polynomial ring over any field k. When k is algebraically closed, a classical result attributed to Castelnuovo establishes an upper bound, which only depends on h, on the number of linearly independent quadrics contained in P. I will overview, in the first half of the talk, standard arguments to obtain upper bounds on number of generators and more generally on graded Betti numbers of homogeneous ideals.

In the second half, which  is based on a joint work with Alessandro De Stefani, I will show how to extend Castelnuovo's result by proving that the number of minimal generators of P in any degree j can be bounded above by an explicit function that only depends on j and h. In addition to providing a bound for generators in any degree j, not just for quadrics, our techniques allow us to drop the assumption that k is algebraically closed. We also obtain analogous upper bounds on  graded Betti numbers of any radical ideal.