Seminar

For any field, we produce an explicit series 4-dimensional domains which are homogeneous graded algebras over the field and whose normalizations (i) have the same degree 1 components as the original algebras, (ii) are generated as algebras in degrees at most 2, yet (iii) admit no uniform upper bound for the discrepancy between the Hilbert functions and the Hilbert polynomials, when viewed as modules over the original homogeneous algebras. The natural habitat of this sort of phenomena is very ample line bundles on projective toric varieties. In the talk, we will overview various related uniform bounds in the homological and combinatorial theory of the rings of sections of such line bundles, some known to exist and some conjectured. Most of these results are joint works with Bruns, Trung, Beck, and Delgado.