Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2206.04856

On the set of Chern numbers in local rings

Hoang Le Truong·Hoang Ngoc Yen
View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

math.AC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Hoang Le Truong, Hoang Ngoc Yen. 2022-06-10. On the set of Chern numbers in local rings. https://arxiv.org/abs/2206.04856

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On triviality of $\mathbb{A}^2$-forms admitting a nontrivial $\mathbb{G}_a$-action

T. Kambayashi had shown that $\mathbb{A}^2$-forms over separable field extensions are necessarily polynomial rings. However, there exist inseparable $\mathbb{A}^2$-forms which are not necessarily polynomial rings. In this paper, we give a structure theorem for $\mathbb{A}^2$-forms over arbitrary field extensions admitting a nontrivial $\mathbb{G}_a$-action. From this structure theorem we derive some conditions under which an $\mathbb{A}^2$-form becomes trivial. In particular, we prove that over a field $k$, a factorial $\mathbb{A}^2$-form having a $k$-rational point and a non-trivial $\mathbb{G}_a$-action is trivial and we also give examples demonstrating that none of these hypotheses can be discarded. As a consequence of the structure theorem, we obtain a generalization of the Zariski Cancellation Theorem for the affine plane over an arbitrary field.

math.AC↗
arXiv2026

Symbolic powers of the ideal of$n$ general points in $P^{n-1}$

Problem L of Fröberg--Lundqvist--Oneto--Shapiro asks for the difference between the Hilbert series of ordinary and symbolic powers of the ideal of general points in projective space. We solve this completely for \(n\) general points of \(\PP^{n-1}\). Besides a closed formula for \[ \HS(S/I^m)-\HS(S/I^{(m)}), \] we determine all minimal monomial generators of \(I^{(m)}\), and describe the symbolic Rees algebra. We also show that containment \(I^{(m)}\subseteq I^r\) is detected solely by initial degrees. This gives the exact containment threshold, the Waldschmidt constant \(\walpha\), the resurgence \(\Res\), and the asymptotic resurgence \(\aRes\): \[ \walpha(I)=\frac{n}{n-1}, \qquad \Res(I)=\aRes(I)=\frac{2(n-1)}{n}. \] We also take the first step beyond \(n\) points: for \(n+1\) general points of \(\PP^{n-1}\) --- again a rigid, non-monomial configuration --- we identify the defining quadrics, resolve the case \(n=3\) completely (a complete intersection, with \(J^{(m)}=J^m\) for all \(m\) and resurgence \(1\)), and propose an exact Waldschmidt-constant formula \(\walpha=\frac{n+1}{n-1}\) for all \(n\), verified computationally in every case we could check.

math.AC↗