0. Introduction (a) Statement of the problem and of the main theorem; some references 233 (b) Corollaries of the main theorem 236 (c) Role of the Brill-Noether matrix 238 (d) Heuristic reasoning for the dimension count; the CastelnuovoSeveri-Kleiman conjecture 240 (e) Notations and terminology 242 1. Reduction of the dimension count to the conjecture (a) Heuristic discussion 244 (b) Geometry of...

We give two kinds of bounds for the Castelnuovo-Mumford regularity of the canonical module and the deficiency modules of a ring, respectively in terms of the homological degree and the Castelnuovo-Mumford regularity of the original ring.

We study the Betti numbers of binomial edge ideal associated to some classes of graphs with large Castelnuovo-Mumford regularity. As an application we give several lower bounds of the Castelnuovo-Mumford regularity of arbitrary graphs depending on induced subgraphs.

A priori bounds of Castelnuovo type for cohomological Hilbert functions.

Bounds for the Castelnuovo-Mumford regularity of Ext modules, over a polynomial ring over a field, are given in terms of the initial degrees, Castelnuovo-Mumford regularities and number of generators of the two graded modules involved. These general bounds are refined in the case the second module is the ring. Other estimates, for instance on the size of graded pieces of these modules, are give...

for every i. One can remark that for the first time, these inequalities were indicated by Margherita Piazzola-Beloch in [1]. She was a pupil of G. Castelnuovo, her paper presents the text of her thesis , G. Castelnuovo was the adviser of the thesis. Thus all (including G. Fano ) the subsequent authors of the variants or generalizations of the Fano inequality are out of the historical responsibi...

1 Veronese rings, Segre embeddings or more generally Segre-Veronese embeddings are very important rings in Algebraic Geometry. In this paper we present an original, elementary way to compute the Hilbert-Poincare series of these rings, as a consequence we compute their Castelnuovo-Mumford regularity, and also the leading term of the h−vector. Moreover, we can compute the Castelnuovo-Mumford regu...