Affine curves in characteristic zero are not always set theoretic complete intersections

Complete intersection monomial curves and the Cohen-Macaulayness of their tangent cones

Let $C({\bf n})$ be a complete intersection monomial curve in the 4-dimensional affine space. In this paper we study the complete intersection property of the monomial curve $C({\bf n}+w{\bf v})$, where $w>0$ is an integer and ${\bf v} \in \mathbb{N}^{4}$. Also we investigate the Cohen-Macaulayness of the tangent cone of $C({\bf n}+w{\bf v})$.

Todd classes of affine cones of Grassmannians

A local ring R is said to be a Roberts ring if tau_R([R]) = [Spec R]_dim R, where tau_R is the Riemann-Roch map for Spec R. Such rings satisfy a vanishing theorem for the Serre intersection multiplicity, as was established by Paul Roberts in his proof of the Serre vanishing conjecture. It is known that complete intersections are Roberts rings, and the first author proved that a determinantal ring is a Roberts ring precisely if it is complete intersection. Let A_d(n) denote the affine cone of the Grassmannian G_d(n) under the Plucker embedding. We determine which of the rings A_d(n) are Roberts…

Complete intersections in certain affine and projective monomial curves

Let $k$ be an arbitrary field, the purpose of this work is to provide families of positive integers $\mathcal{A} = \{d_1,\ldots,d_n\}$ such that either the toric ideal $I_{\mathcal A}$ of the affine monomial curve $\mathcal C = \{(t^{d_1},\ldots,\,t^{d_n}) \ | \ t \in k\} \subset \mathbb{A}_k^n$ or the toric ideal $I_{\mathcal A^{\star}}$ of its projective closure ${\mathcal C^{\star}} \subset \mathbb{P}_k^n$ is a complete intersection. More precisely, we characterize the complete intersection property for $I_{\mathcal A}$ and for $I_{\mathcal A^{\star}}$ when: (a) $\mathcal{A}$ is a generaliz…

Algorithms for Checking Zero-Dimensional Complete Intersections

Given a 0-dimensional affine K-algebra R=K[x_1,...,x_n]/I, where I is an ideal in a polynomial ring K[x_1,...,x_n] over a field K, or, equivalently, given a 0-dimensional affine scheme, we construct effective algorithms for checking whether R is a complete intersection at a maximal ideal, whether R is locally a complete intersection, and whether R is a strict complete intersection. These algorithms are based on Wiebe's characterisation of 0-dimensional local complete intersections via the 0-th Fitting ideal of the maximal ideal. They allow us to detect which generators of I form a regular sequ…

Gröbner scheme in the Hilbert scheme and complete intersection monomial ideals

Let $k$ be a commutative ring and $S=k[x_0, \ldots, x_n]$ be a polynomial ring over $k$ with a monomial order. For any monomial ideal $J$, there exists an affine $k$-scheme of finite type, called Gröbner scheme, which parameterizes all homogeneous reduced Gröbner bases in $S$ whose initial ideal is $J$. Here we functorially show that the Gröbner scheme is a locally closed subscheme of the Hilbert scheme if $J$ is a saturated ideal. In the process, we also show that the Gröbner scheme consists of complete intersections if $J$ defines a complete intersection.

Producing Set Theoretic Complete Intersection Monomial Curves in P^n

In this paper we produce infinitely many examples of set-theoretic complete intersection monomial curves in $\mathbb{P}^{n+1}$, starting with a set-theoretic complete intersection monomial curve in $\mathbb{P}^{n}$ . In most of the cases our results cannot be obtained through semigroup gluing technique and we can tell apart explicitly which cases are new.

完全交集在余秩一中的方法

Methods in complete intersections in corank one

本文探讨了在余秩一情况下，如何将完全交集理论的方法应用于稳定自由模的消去定理，指出需要改进以使该方法有效，并提出了相关假设及其后果。

Certain minimal varieties are set-theoretic complete intersections

We present a class of homogeneous ideals which are generated by monomials and binomials of degree two and are set-theoretic complete intersections. This class includes certain reducible varieties of minimal degree and, in particular, the presentation ideals of the fiber cone algebras of monomial varieties of codimension two.

Minimum distance functions of complete intersections

We study the footprint function, with respect to a monomial order, of complete intersection graded ideals in a polynomial ring with coefficients in a field. For graded ideals of dimension one, whose initial ideal is a complete intersection, we give a formula for the footprint function and a sharp lower bound for the corresponding minimum distance function. This allows us to recover a formula for the minimum distance of an affine cartesian code and the fact that in this case the minimum distance and the footprint functions coincide. Then we present an extension of a result of Alon and Füredi, a…

Almost set-theoretic complete intersections in characteristic zero

We present a class of toric varieties $V$ which, over any algebraically closed field of characteristic zero, are defined by codim $V$+1 binomial equations.

某些实光滑仿射三维簇上秩为2的稳定自由模

Stably free modules of rank $2$ over certain real smooth affine threefolds

针对维数3且满足无实极大理想或其实极大理想交集高度≥1的实光滑仿射整环R，证明秩2稳定自由R-模皆自由当且仅当埃尔米特K-理论群W_{SL}(R)平凡。

Equations defining recursive extensions as set theoretic complete intersections

Based on the fact that projective monomial curves in the plane are complete intersections, we give an effective inductive method for creating infinitely many monomial curves in the projective $n$-space that are set theoretic complete intersections. We illustrate our main result by giving different infinite families of examples. Our proof is constructive and provides one binomial and $(n-2)$ polynomial explicit equations for the hypersurfaces cutting out the curve in question.

Counting Solutions to Binomial Complete Intersections

We study the problem of counting the total number of affine solutions of a system of n binomials in n variables over an algebraically closed field of characteristic zero. We show that we may decide in polynomial time if that number is finite. We give a combinatorial formula for computing the total number of affine solutions (with or without multiplicity) from which we deduce that this counting problem is #P-complete. We discuss special cases in which this formula may be computed in polynomial time; in particular, this is true for generic exponent vectors.

${\mathbb P}^1$-gluing for local complete intersections

We prove an analogue of the Affine Horrocks' Theorem for local complete intersection ideals of height $n$ in $R[T]$, where $R$ is a regular domain of dimension $d$, which is essentially of finite type over an infinite perfect field of characteristic unequal to $2$, and $2n\geq d+3$.

On the Hermitian curve and its intersections with some conics

We classify completely the intersections of the Hermitian curve with parabolas in the affine plane. To obtain our results we employ well-known algebraic methods for finite fields and geometric properties of the curve automorphisms. In particular, we provide explicit counting formulas that have also applications to some Hermitian codes.

有限补集单纯仿射半群的Buchsbaum性质

Buchsbaumness of finite complement simplicial affine semigroups

本文分类了补集有限的Buchsbaum单纯仿射半群，证明其Buchsbaum性质等价于间隙集与伪Frobenius元素集相等，给出完整结构及最大嵌入维数下的最小表示基数公式，并指出该性质在粘合下不保持。

The Stanley-Reisner ideals of polygons as set-theoretic complete intersections

We show that the Stanley-Reisner ideal of the one-dimensional simplicial complex whose diagram is an $n$-gon is always a set-theoretic complete intersection in any positive characteristic.

Symbolic Rees algebras and set-theoretic complete intersections

In this paper we extend a result of Cowsik on set-theoretic complete intersection and a result Huneke, Morales and Goto and Nishida about Noetherian symbolic Rees algebras of ideals. As applications, we show that the symbolic Rees algebras of the following ideals are Noetherian and the ideals are set-theoretic complete intersections: (a) the edge ideal of a complete graph, (b) the Fermat ideal and (c) the Jacobian ideal of a certain hyperplane arrangement.

Symbolic powers, set-theoretic complete intersection and certain invariants

In this survey article we give a brief history of symbolic powers and its connection with the interesting problem of set-theoretic complete intersection. We also state a few problems and conjectures. Recently, in connection to symbolic powers is the containment problem. We list a few interesting results and related problems on the resurgence, Waldschmidt constant and Castelnuovo-Mumford regularity.

集合论意义下的完美理想与剩余交

Set-Theoretically Perfect Ideals and Residual Intersections

本文研究满足Serre条件Sₛ环中的代数剩余交，证明其存在自由逼近（即具相同根的完美子理想），由此确定剩余交重数的一致上界；在正特征下，剩余交是上同调完全交，故其簇在余维1连通。

Complete intersection ideals and a question of Nori

This is my PhD thesis from 2004 under Prof. S.M. Bhatwadekar. Here we answer a question of Nori and prove the following result. Let $A$ be a smooth affine domain of dimension $d$ over an infinite perfect field. Let $I$ be an ideal of $A[T]$ of height $n$ such that $2n \geq d+3$. Given surjections $ϕ:(A[T]/I)^n \rightarrow I/I^2$ and $ρ:A^n \rightarrow I(0)$ such that $ϕ(0)=ρ\otimes A/I(0)$, then there exist a surjection $Φ:A[T]^n \rightarrow I$ such that $Φ(0)=ρ$ and $Φ\otimes A/I =ϕ$. This is a joint work with S.M. Bhatwadekar.