Jan Snellman
We study two partial orders on $[x_1,...,x_n]$, the free abelian monoid on ${x_1,...,x_n}$. These partial orders, which we call the ``strongly stable'' and the ``stable'' partial order, are defined by the property that their filters are precisely the strongly stable and the stable monoid ideals. These ideals arise in the study of generic initial ideals.
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Partial Betti splittings with applications to binomial edge ideals
本文提出齐次理想的偏 Betti 分裂概念，并将其用于二项式边理想，推广割边相关结果、给出通用分裂描述且算出树情形的总第二 Betti 数。
We construct a noncoherent initial ideal of an ideal in the exterior algebra of order 6, answering a question of D. Maclagan (2000). We also give a method for constructing noncoherent initial ideals in exterior algebras using certain noncoherent term orders.
This is a survey article on Gorenstein initial complexes of extensively studied ideals in commutative algebra and algebraic geometry. These include defining ideals of Segre and Veronese varieties, toric deformations of flag varieties known as Hibi ideals, determinantal ideals of generic matrices of indeterminates, and ideals generated by Pfaffians of generic skew symmetric matrices. We give a summary of recent work on the construction of squarefree Gorenstein initial ideals of these ideals when the ideals are themselves Gorenstein. We also present our own independent results for the Segre, Ver…
We will study monomial ideals $I$ in the exterior algebra as well as in the polynomial ring whose generic initial ideal is constant for all term orders up to permutations of variables. First, in the exterior algebra, we determine all graphs and all flag complexes whose exterior face ideal satisfies the above condition. Second, in the polynomial ring, it will be shown that the generic initial ideal $\gin_σ(I(G))$ of the edge ideal $I(G)$ of a graph $G$ is constant for all term orders $σ$ up to permutations of variables if and only if $G$ is a complete bipartite graph.
On the gradient of a monomial ideal
本文研究了单项式理想与其梯度理想之间的正则度关系，证明了即使对于具有线性解析的单项式理想，正则度仍不可比较，并引入了具有微分线性解析的单项式理想类别。
In 1988 Kalai construct a large class of simplicial spheres, called squeezed spheres, and in 1991 presented a conjectured about generic initial ideals of Stanley--Reisner ideals of squeezed spheres. In the present paper this conjecture will be proved. In order to prove Kalai's conjecture, based on the fact that every squeezed $(d-1)$-sphere is the boundary of a certain $d$-ball, called a squeezed $d$-ball, generic initial ideals of Stanley--Reisner ideals of squeezed balls will be determined. In addition, generic initial ideals of exterior face ideals of squeezed balls are determined. On the o…
We determine a term order on the monomials in the variables $\varx{i}{j}$, $1 \leq i < j \leq n$, such that corresponding initial ideal of the ideal of Pfaffians of degree $r$ of a generic $n$ by $n$ skew-symmetric matrix is the Stanley-Reisner ideal of a join of a simplicial sphere and a simplex. Moreover, we demonstrate that the Pfaffians of the $2r$ by $2r$ skew-symmetric submatrices form a Gröbner basis for the given term order. The same methods and similar term orders as for the Pfaffians also yield squarefree initial ideals for certain determinantal ideals. Yet, in contrast to the case o…
We study the behavior of generic initial ideals with respect to fibre products. In our main result we determine the generic initial ideal of the fibre product with respect to the reverse lexicographic order. As an application we compute the symmetric algebraic shifted complex of two disjoint simplicial complexes as was conjectured by Kalai. This result is the symmetric analogue of a theorem of Nevo who determined the exterior algebraic shifted complex of two disjoint simplicial complexes as predicted by Kalai.
A Structural Property of Generic Initial Ideals
该研究证明一般初始理想的渐近结构性质，为Hilbert函数等多个代数相关领域的问题提供了新结果。
We compute the initial ideals, with respect to certain conveniently chosen term orders, of ideals of tangent cones at torus fixed points to Schubert varieties in orthogonal Grassmannians. The initial ideals turn out to be square-free monomial ideals and therefore Stanley-Reisner face rings of simplicial complexes. We describe these complexes. The maximal faces of these complexes encode certain sets of non-intersecting lattice paths.
A normal (respectively, graded normal) vector configuration $A$ defines the toric ideal $I_A$ of a normal (respectively, projectively normal) toric variety. These ideals are Cohen-Macaulay, and when $A$ is normal and graded, $I_A$ is generated in degree at most the dimension of $I_A$. Based on this, Sturmfels asked if these properties extend to initial ideals -- when $A$ is normal, is there an initial ideal of $I_A$ that is Cohen-Macaulay, and when $A$ is normal and graded, does $I_A$ have a Gröbner basis generated in degree at most $dim(I_A)$ ? In this paper, we answer both questions positive…
In this paper, we study the sequences with (non-consecutive) two zero-symbols and ideal autocorrelation, which are also known as almost $m$-ary nearly perfect sequences. We show that these sequences are equivalent to $\ell$-partial direct product difference sets (PDPDS), then we extend known results on the sequences with two consecutive zero-symbols to non-consecutive case. Next, we study the notion of multipliers and orbit combination for $\ell$-PDPDS. Finally, we present a construction method for a family of almost quaternary sequences with ideal autocorrelation by using cyclotomic classes.
Multi-graded generic initial ideals, regularity, and the optimal colorful fractional Helly theorem for $d$-Leray complexes
该研究针对多级通用初始理想的正则性问题，基于特定单项式序证明了R/I正则性的下界，并将其应用得出d-Leray单纯复形的最优彩色分数Helly定理。
What kind of reduced monomial schemes can be obtained as a Gröbner degeneration of a smooth projective variety? Our conjectured answer is: only Stanley-Reisner schemes associated to acyclic Cohen-Macaulay simplicial complexes. This would imply, in particular, that only curves of genus zero have such a degeneration. We prove this conjecture for degrevlex orders, for elliptic curves over real number fields, for boundaries of cross-polytopes, and for leafless graphs. We discuss consequences for rational and F-rational singularities of algebras with straightening laws.
Let $I_n$ be the ideal of all algebraic relations on the slopes of the $\binom{n}{2}$ lines formed by placing $n$ points in a plane and connecting each pair of points with a line. Under each of two natural term orders, the initial ideal of $I_n$ is generated by monomials corresponding to permutations satisfying a certain pattern-avoidance condition. We show bijectively that these permutations are enumerated by the updown (or Euler) numbers, thereby obtaining a formula for the number of generators of the initial ideal of $I_n$ in each degree.
The order polytope $\mathcal{O}(P)$ and the chain polytope $\mathcal{C}(P)$ associated to a partially ordered set $P$ are studied. In this paper, we introduce the convex polytope $Γ(\mathcal{O}(P), -\mathcal{C}(Q))$ which is the convex hull of $\mathcal{O}(P) \cup (-\mathcal{C}(Q))$, where both $P$ and $Q$ are partially ordered sets with $|P|=|Q|=d$. It will be shown that $Γ(\mathcal{O}(P), -\mathcal{C}(Q))$ is a normal and Gorenstein Fano polytope by using the theory of reverse lexicographic squarefree initial ideals of toric ideals.
Initial ideals of tangent cones to Richardson varieties in the Symplectic Grassmannian
本文研究辛格拉斯曼流形中 Richardson 簇切锥的初始理想，构造其任意 T-不动点处切锥理想的显式 Gröbner 基，推广了已有结果。
In attempting to understand how combinatorial modifications alter algebraic properties of monomial ideals, several authors have investigated the process of adding "whiskers" to graphs. In this paper, we study a similar construction to build a simplicial complex $Δ_χ$ from a coloring $χ$ of a subset of the vertices of $Δ$, and give necessary and sufficient conditions for this construction to produce vertex decomposable simplicial complexes. We apply this work to strengthen and give new proofs about sequentially Cohen-Macaulay edge ideals of graphs.
Let $G$ be a cycle or a complete bipartite graph. We show that the binomial edge ideal $J_{G}$ and its initial ideal with respect to the lexicographic order have the same extremal Betti number.
A Richardson variety $X_\ga^\gc$ in the Orthogonal Grassmannian is defined to be the intersection of a Schubert variety $X^\gc$ in the Orthogonal Grassmannian and a opposite Schubert variety $X_\ga$ therein. We give an explicit description of the initial ideal (with respect to certain conveniently chosen term order) for the ideal of the tangent cone at any $T$-fixed point of $X_\ga^\gc$, thus generalizing a result of Raghavan-Upadhyay \cite{Ra-Up2}. Our proof is based on a generalization of the Robinson-Schensted-Knuth (RSK) correspondence, which we call the Orthogonal bounded RSK (OBRSK). The…
We study the partial Brauer monoid and its planar submonoid, the Motzkin monoid. We conduct a thorough investigation of the structure of both monoids, providing information on normal forms, Green's relations, regularity, ideals, idempotent generation, minimal (idempotent) generating sets, and so on. We obtain necessary and sufficient conditions under which the ideals of these monoids are idempotent-generated. We find formulae for the rank (smallest size of a generating set) of each ideal, and for the idempotent rank (smallest size of an idempotent generating set) of the idempotent-generated su…
Bounded powers of edge ideals: Pseudo-Gorenstein and Level polytopes
本文研究了有限次边理想离散多项组所生成的level*和伪克罗内克*多面体的性质与特征。
The construction of joins and secant varieties is studied in the combinatorial context of monomial ideals. For ideals generated by quadratic monomials, the generators of the secant ideals are obstructions to graph colorings, and this leads to a commutative algebra version of the Strong Perfect Graph Theorem. Given any projective variety and any term order, we explore whether the initial ideal of the secant ideal coincides with the secant ideal of the initial ideal. For toric varieties, this leads to the notion of delightful triangulations of convex polytopes.
Regular subdivisions, bounds on initial ideals, and categorical limits
该研究提出通用框架，将点构型与齐次理想关联，用正则细分定义初始理想的上下界并通过范畴（余）极限解释，还证明精确界对应权重构成二级扇的子扇，推广了环面簇与Grassmann簇的相关构造。
A partial lattice P is ideal-projective, with respect to a class C of lattices, if for every K $\in$ C and every homomorphism $ϕ$ of partial lattices from P to the ideal lattice of K, there are arbitrarily large choice functions f : P $\rightarrow$ K for $ϕ$ that are also homomorphisms of partial lattices. This extends the traditional concept of (sharp) transferability of a lattice with respect to C. We prove the following: (1) A finite lattice P, belonging to a variety V, is sharply transferable with respect to V iff it is projective with respect to V and weakly distributive lattice homomorph…
We explore the consequences of an ideal I of real polynomials having a real radical initial ideal, both for the geometry of the real variety of I and as an application to sums of squares representations of polynomials. We show that if in_w(I) is real radical for a vector w in the tropical variety, then w is in the logarithmic set of the real variety. We also give algebraic sufficient conditions for w to be in the logarithmic limit set of a more general semialgebraic set. If in addition the entries of w are positive, then the corresponding quadratic module is stable. In particular, if in_w(I) i…
The arc complexes of partially decorated hyperbolic polygons
该研究针对部分装饰的理想与理想一次穿孔双曲多边形，在组合框架下证明其弧复形为闭分段线性球，并借此实现这类多边形弱加长形变的参数化。
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