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Hoang Le Truong

Publications and source records attributed to Hoang Le Truong.

At least 19 recordsLinked to original sources
arXiv2026

Density functions for filtrations of graded ideals

We study density functions associated with filtrations of graded ideals in standard graded domains. Extending earlier work, we develop a general theory for arbitrary filtrations and establish their fundamental properties, including existence, continuity, log-concavity, and integral representations. We show that every density function is locally uniformly approximable by piecewise polynomial density functions arising from truncated Noetherian filtrations. We prove differentiability properties of saturation density functions and give a numerical characterization of when the symbolic powers of a graded radical ideal in a polynomial ring coincide with the integral closures of its ordinary powers. Finally, we establish several inequalities for the mixed multiplicities of equigenerated ideals.

math.AC↗
arXiv2024

Extensions and paracanonical curves of genus 6

In this note we give a computationally easy to use method to compute a maximal extension of certain varieties. As a application we prove that a general paracanonical curve C genus 6 as a codimension three subvarieties of P^4 extend to precisely 26 families of surfaces Y in P^5.

math.AG↗
arXiv2022

Flexibility of Affine cones over Mukai fourfolds of genus $g\ge7$

We show that the affine cones over a general Fano-Mukai fourfold of genus $g=7$, $8$ and $9$ are flexible. Equivalently, there is an infinitely transitive action of the special automorphism group on such affine cones. In particular, any Mukai fourfold of genus $7,8$ and $9$ is $\Bbb A^2$-cylindrical.

math.AG↗
arXiv2022

On the set of Chern numbers in local rings

This paper purposes to characterize Noetherian local rings $(R, \mathfrak{m})$ such that the Chern numbers of certain $\mathfrak{m}$-primary ideals in $R$ bounded above or range among only finitely many values. Consequently, we characterize the Gorensteinness, Cohen-Macaulayness, generalized Cohen-Macaulayness of local rings in terms of the behavior of its Chern numbers.

math.AC↗
arXiv2022

On the sectional genera and Cohen-Macaulay rings

We explore the behavior of the sectional genera of certain primary ideals in Noetherian local rings. In this paper, we provide characterizations of a Cohen-Macaulay local ring in terms of the sectional genera, the Cohen-Macaulay type, and the second Hilbert coefficients for certain primary ideals. We also characterize Gorenstein rings and quasi-Buchsbaum rings in terms of the sectional genera for certain primary ideals.

math.AC↗
arXiv2022

Classification and geometric properties of surfaces with property ${\bf N}_{3,3}$

Let $X$ be a closed subscheme of codimension $e$ in a projective space. One says that $X$ satisfies property ${\bf N}_{d,p}$, if the $i$-th syzygies of the homogeneous coordinate ring are generated by elements of degree $<d+i$ for $0\le i\le p$. The geometric and algebraic properties of smooth projective varieties satisfying property ${\bf N}_{2,e}$ are well understood, and the complete classification of these varieties is a classical result. The aim of this paper is to study the next case: projective surfaces in $\Bbb P^5$ satisfying property ${\bf N}_{3,3}$. In particular, we give a classification of such varieties using adjunction mappings and we also provide illuminating examples of our results via calculations done with Macaulay 2. As corollaries, we study the CI-biliaison equivalence class of smooth projective surfaces of degree $10$ satisfying property ${\bf N}_{3,3}$ on a cubic fourfold.

math.AG↗
arXiv2022

Stable Ulrich bundles on cubic fourfolds

In this paper, we give necessary and sufficient conditions for the existence of Ulrich bundles on cubic fourfold $X$ of given rank $r$. As consequences, we show that for every integer $r\ge 2$ there exists a family of indecomposable rank $r$ Ulrich bundles on the certain cubic fourfolds, depending roughly on $r$ parameters, and in particular they are of wild representation type; special surfaces on the cubic fourfolds are explicitly constructed by Macaulay2; a new $19$-dimensional family of projective ten-dimensional irreducible holomorphic symplectic manifolds associated to a certain cubic fourfold is constructed; and for certain cubic fourfold $X$, there exist arithmetically Cohen-Macaulay smooth surface $Y \subset X$ which are not an intersection $X \cap T$ for a codimension two subvariety $T \subset \Bbb P^5$.

math.AG↗
arXiv2021

Canonical stretched rings

In this paper, we introduce the concept of canonical stretched rings, sparse stretched rings and maximum sparse ideals. Then we give characterizations of canonical stretched rings and sparse stretched rings; and a characterization of Gorenstein rings in terms of their maximum sparse ideals. Several explicit examples are provided along the paper to illustrate such rings.

math.AC↗
arXiv2021

A note on Chern coefficients and Cohen-Macaulay rings

In this paper, we investigate the relationship between the index of reducibility and Chern coefficients for primary ideals. As an application, we give characterizations of a Cohen-Macaulay ring in terms of its type, irreducible multiplicity, and Chern coefficients with respect to certain parameter ideals in Noetherian local rings.

math.AC↗
arXiv2021

On Hilbert coefficients and sequentially Cohen-Macaulay rings

In this paper, we explore the relation between the index of reducibility and the Hilbert coefficients in local rings. Consequently, the main result of this study provides a characterization of a sequentially Cohen-Macaulay ring in terms of its Hilbert coefficients for non-parameter ideals. As corollaries to the main theorem, we obtain characterizations of a Gorenstein/Cohen-Macaulay ring in terms of its Chern coefficients for parameter ideals.

math.AC↗
arXiv2019

On the structure of the Sally module and the second normal Hilbert coefficient

The Hilbert coefficients of the normal filtration give important geometric information on the base ring like the pseudo-rationality. The Sally module was introduced by W.V. Vasconcelos and it is useful to connect the Hilbert coefficients to the homological properties of the associated graded module of a Noetherian filtration. In this paper we give a complete structure of the Sally module in the case the second normal Hilbert coefficient attains almost minimal value in an analytically unramified Cohen-Macaulay local ring. As a consequence, in this case we present a complete description of the Hilbert function of the associated graded ring of the normal filtration. A deep analysis of the vanishing of the third Hilbert coefficient has been necessary. This study is related to a long-standing conjecture stated by S. Itoh.

math.AC↗
arXiv2016

When are the Rees algebras of parameter ideals almost Gorenstein graded rings?

Let $A$ be a Cohen-Macaulay local ring with $\operatorname{dim} A = d\ge 3$, possessing the canonical module ${\mathrm K}_A$. Let $a_1, a_2, \ldots, a_r$ $(3 \le r \le d)$ be a subsystem of parameters of $A$ and set $Q= (a_1, a_2, \ldots, a_r)$. It is shown that if the Rees algebra ${\mathcal R}(Q)$ of $Q$ is an almost Gorenstein graded ring, then $A$ is a regular local ring and $a_1, a_2, \ldots, a_r$ is a part of a regular system of parameters of $A$.

math.AC↗
arXiv2015

Hilbert functions of socle ideals

In this paper, we explore a relationship between Hilbert functions and the irreducible decompositions of ideals in local rings. Applications are given to characterize the regularity, Gorensteinness, Cohen-Macaulayness and sequentially Cohen-Macaulayness of local rings.

math.AC↗