Multigraded regularity, a*-invariant and the minimal free resolution

In recent years, two different multigraded variants of Castelnuovo-Mumford regularity have been developed, namely multigraded regularity, defined by the vanishing of multigraded pieces of local cohomology modules, and the resolution regularity vector, defined by the multidegrees in a minimal free resolution. In this paper, we study the relationship between multigraded regularity and the resolution regularity vector. Our method is to investigate multigraded variants of the usual a*-invariant. This, in particular, provides an effective approach to examining the vanishing of multigraded pieces of local cohomology modules with respect to different graded irrelevant ideals.Comment: Final version to appear in J. Algebra; 24 page

Asymptotic linearity of regularity and a*-invariant of powers of ideals

Let X = Proj R be a projective scheme over a field k, and let I be an ideal in R generated by forms of the same degree d. Let Y --> X be the blowing up of X along the subscheme defined by I, and let f: Y --> Z be the projection of Y given by the divisor dH - E, where E is the exceptional divisor of the blowup and H is the pullback of a general hyperplane in X. We investigate how the asymptotic linearity of the regularity and a*-invariant of I^q (for q large) is related to invariants of fibers of f.Comment: 11 pages, revision: get rid of the condition that R is a polynomial ring in the last theorem

Box-shaped matrices and the defining ideal of certain blowup surfaces

We study the defining equations of projective embeddings of the blowup of P^2 at a set of {d+1 \choose 2} number of points in generic position. To do this, we first generalize the notion of a matrix, its ideal of 2x2 minors to that of a box-shaped matrix. Our work completes previous works of Geramita and Gimigliano

Minimal free resolutions and asymptotic behavior of multigraded regularity

Let S be a standard N^k-graded polynomial ring over a field. Let I be a multigraded homogeneous ideal in S and let M be a finitely generated Z^k-graded S-module. We prove that the resolution regularity, a multigraded variant of Castelnuovo-Mumford regularity, of I^nM is asymptotically a linear function. This shows that the well known Z-graded phenomenon carries to multigraded situation.Comment: Final version to appear in J. Algebra; 18 page

SINO-VIETNAMESE PEOPLE’S ECONOMIC ACTIVITIES IN DISTRICTS 5, 6, AND 11 OF HO CHI MINH CITY, VIETNAM (1996-2016)

This article clarifies the involvement of the Sino-Vietnamese people in Ho Chi Minh City’s economy from 1996 to 2016. Vietnam has accelerated the process of industrialization and modernization since 1996, and Ho Chi Minh City now has a thriving local economy. As with other ethnic groups, the Sino-Vietnamese people have made a considerable contribution to boosting the local economy. Based on the theory of functionalism, culture-economy relationships, and primary anthropological research methods, this article assesses the economic contributions of the Sino-Vietnamese in industry and handicrafts, commerce and services, and finance and credit. However, there are many obstacles for the Sino-Vietnamese economy. The quality of goods is insufficient to meet consumer satisfaction. The harshly competitive market confronts their businesses with numerous challenges. The abandonment of traditional professions by the young Sino-Vietnamese generation poses a threat to traditional businesses, and COVID-19 is a pressing issue for their economy. This article proposes realistic solutions to encourage the Sino-Vietnamese people to overcome disadvantages and contribute to the economy of Ho Chi Minh City in the future

Embedded Associated Primes of Powers of Square-free Monomial Ideals

An ideal I in a Noetherian ring R is normally torsion-free if Ass(R/I^t)=Ass(R/I) for all natural numbers t. We develop a technique to inductively study normally torsion-free square-free monomial ideals. In particular, we show that if a square-free monomial ideal I is minimally not normally torsion-free then the least power t such that I^t has embedded primes is bigger than beta_1, where beta_1 is the monomial grade of I, which is equal to the matching number of the hypergraph H(I) associated to I. If in addition I fails to have the packing property, then embedded primes of I^t do occur when t=beta_1 +1. As an application, we investigate how these results relate to a conjecture of Conforti and Cornu\'ejols.Comment: 15 pages, changes have been made to the title, introduction, and background material, and an example has been added. To appear in JPA