Ze Min Zeng
It is proved in this paper that a locally complete intersection curve in a smooth affine C-algebra with trival conormal bundle is a set theoretic complete intersection if its corresponding class in the Grothendieck Group is torsion.
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On complete intersection projective closures of monomial curves
研究四维仿射空间中单项式曲线射影闭包的完全交性质，给出其定义理想为完全交的刻画，还探讨其是否算术科恩 - 麦考利。
On set-theoretic complete intersections for smooth curves in three-dimensional affine schemes
本文证明了三维诺特环仿射概形中的局部完全交曲线是集合论完全交，并将类似结果推广到四维仿射代数中的曲面及平凡余法丛情形。
For any affine-variety code we show how to construct an ideal whose solutions correspond to codewords with any assigned weight. We classify completely the intersections of the Hermitian curve with lines and parabolas (in the $\mathbb{F}_{q^2}$ affine plane). Starting from both results, we are able to obtain geometric characterizations for small-weight codewords for some families of Hermitian codes over any $\mathbb{F}_{q^2}$. From the geometric characterization, we obtain explicit formulae. In particular, we determine the number of minimum-weight codewords for all Hermitian codes with $d\leq q…
We describe a class of affine toric varieties $V$ that are set-theoretically minimally defined by codim $V+1$ binomial equations over fields of any characteristic.
Using aritmethic conditions on affine semigroups we prove that for a simplicial toric variety of codimension 2 the property of being a set-theoretic complete intersection on binomials in characteristic $p$ holds either for all primes $p$, or for no prime $p$, or for exactly one prime $p$.
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 $C({\bf n}+w{\bf v})$.
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…
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…
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…
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.
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.
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。
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…
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.
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.
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。
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.
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.
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 $2n\geq d+3$.
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.
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。
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.
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.
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.
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。
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.
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