We establish the first previously unknown case of the EisenbudHarris conjecture in Castelnuovo theory concerning algebraic curves of high genus in Pn. The problem is reduced to a question about zero-dimensional schemes Γ ⊂ P in symmetric position with certain constrains on the Hilbert function. The method of Gröbner bases is then applied to study the homogeneous ideal of Γ.

We give characterizations of contractible curves on proper normal algebraic surfaces in terms of complementary Weil divisors. From this we obtain some generalizations of the classical criteria for contractibility of Castelnuovo and Artin. Furthermore, we will derive a finiteness result on homogeneous spectra defined by Weil divisors on proper normal algebraic surfaces.

D. Bayer and M. Stillman showed that Gröbner bases can be used to compute the Castelnuovo-Mumford regularity which is a measure for the vanishing of graded local cohomology modules. The aim of this paper is to show that the same method can be applied to study other cohomological invariants as well as the reduction number.

We study the initial ideal of binomial edge ideal in degree 2 ([in<(JG)]2), associated to a graph G. We computed dimension, depth, Castelnuovo-Mumford regularity, Hilbert function and Betti numbers of [in<(JG)]2 for some classes of graphs. AMS Mathematics Subject Classification (2010): 05E40, 16E30

In this paper we give a nice formula for the CastelnuovoMumford regularity of the Segre product of modules, under some suitable hypotheses. This extends recent results of David A. Cox, and Evgeny Materov (2009).

In this paper we study and relate several invariants connected to the solving degree of a polynomial system. This provides rigorous framework for estimating complexity system equations via Gröbner bases methods. Our main results include connection between last fall one regularity Castelnuovo–Mumford regularity.

T he golden age of the Italian school of algebraic geometry began with Antonio Luigi GaudenzioGiuseppeCremona and included among its main contributors Enrico Castelnuovo, Federigo Enriques, and Francesco Severi. The Italian school spanned nearly a century, from the unification of Italy in 1861 to Enriques’s posthumously published postWorld War II monograph on algebraic surfaces [Enrq 49]. In th...