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In this paper, we study a family of binomial ideals defining monomial curves in the n-dimensional affine space determined by n hypersurfaces of the form $x_i^{c_i} - x_1^{u_{i1}} \cdots x_n^{u_{1n}}$ in $\Bbbk [x_1, \ldots , x_n]$ with $u_{ii} = 0, \ i\in \{ 1, \ldots , n\}$. We prove that the monomial curves in that family are set-theoretic complete intersections. Moreover, if the monomial curve is irreducible, we compute some invariants such as genus, type and Frobenius number of the corresponding numerical semigroup. We also describe a method to produce set-theoretic complete intersection semigroup ideals of arbitrary large height. [ABSTRACT FROM AUTHOR]

Geometric vertex decomposition and liaison are two frameworks that have been used to produce similar results about similar families of algebraic varieties. In this paper, we establish an explicit connection between these approaches. In particular, we show that each geometrically vertex decomposable ideal is linked by a sequence of elementary G-biliaisons of height $1$ to an ideal of indeterminates and, conversely, that every G-biliaison of a certain type gives rise to a geometric vertex decomposition. As a consequence, we can immediately conclude that several well-known families of ideals are glicci, including Schubert determinantal ideals, defining ideals of varieties of complexes and defining ideals of graded lower bound cluster algebras. [ABSTRACT FROM AUTHOR]

Let IX be the saturated homogeneous ideal defining a codimension two arithmetically Cohen-Macaulay scheme X ⊆ P n , and let I X (m) denote its m-th symbolic power. We are interested in when I X (m) = I X m. We survey what is known about this problem when X is locally a complete intersection, and in particular, we review the classification of when I X (m) = I X m for all m ≥ 1. We then discuss how one might weaken these hypotheses, but still obtain equality between the symbolic and ordinary powers. Finally, we show that this classification allows one to: (1) simplify known results about symbolic powers of ideals of points in P 1 × P 1 ; (2) verify a conjecture of Guardo, Harbourne, and Van Tuyl, and (3) provide additional evidence to a conjecture of Römer. [ABSTRACT FROM AUTHOR]

Let R = k [ x 1 , ... , x n ] be the polynomial ring in n variables over a field k and let I be a matroidal ideal of degree d. In this paper, we study the unmixedness properties and the arithmetical rank of I. Moreover, we show that a r a (I) = n - d + 1 . This answers the conjecture made by Chiang-Hsieh (Comm Algebra 38:944–952, 2010, Conjecture). [ABSTRACT FROM AUTHOR]

We study Gorenstein liaison of codimension two subschemes of anarithmetically Gorenstein scheme X. Our main result is a criterion for two suchsubschemes to be in the same Gorenstein liaison class, in terms of the categoryof ACM sheaves on X. As a consequence we obtain a criterion for X to have theproperty that every codimension 2 arithmetically Cohen-Macaulay subscheme is inthe Gorenstein liaison class of a complete intersection. Using these tools weprove that every arithmetically Gorenstein subscheme of $\mathbb{P}^n$ is inthe Gorenstein liaison class of a complete intesection and we are able tocharacterize the Gorenstein liaison classes of curves on a nonsingular quadricthreefold in $\mathbb{P}^4$.

Abstract: We study divided power structures on finitely generated k‐algebras, where k is a field of positive characteristic p. As an application we show examples of zero‐dimensional Gorenstein k‐schemes that do not lift to a fixed noetherian local ring of non‐equal characteristic. We also show that Frobenius neighbourhoods of a singular point of a general hypersurface of large dimension have no liftings to mildly ramified rings of non‐equal characteristic. [ABSTRACT FROM AUTHOR]

Abstract: Homological index of a holomorphic 1‐form on a complex‐analytic variety with an isolated singular point is an analogue of the usual index of a 1‐form on a non‐singular manifold. One can say that it corresponds to the top Chern number of a manifold. We offer a definition of homological indices for collections of 1‐forms on a (purely dimensional) complex‐analytic variety with an isolated singular point corresponding to other Chern numbers. We also define new invariants of germs of complex‐analytic varieties with isolated singular points related to ‘vanishing Chern numbers’ at them. [ABSTRACT FROM AUTHOR]

Let $$L\subset \mathbb {Z}^n$$ be a lattice and $$I_L=\langle x^{\mathbf {u}}-x^{\mathbf {v}}:\ {\mathbf {u}}-{\mathbf {v}}\in L\rangle $$ be the corresponding lattice ideal in $$\Bbbk [x_1,\ldots , x_n]$$ , where $$\Bbbk $$ is a field. In this paper we describe minimal binomial generating sets of $$I_L$$ and their invariants. We use as a main tool a graph construction on equivalence classes of fibers of $$I_L$$ . As one application of the theory developed we characterize binomial complete intersection lattice ideals, a longstanding open problem in the case of non-positive lattices. [ABSTRACT FROM AUTHOR]

Some relations between cohomological dimensions and depths of linked ideals are investigated and discussed by various examples. [ABSTRACT FROM PUBLISHER]

Given a 0-dimensional subscheme $${\mathbb X}$$ of a projective space $${\mathbb P}^n_K$$ over a field K, we characterize in different ways whether $${\mathbb X}$$ is the complete intersection of n hypersurfaces. Besides a generalization of the notion of a Cayley-Bacharach scheme, these characterizations involve the Kähler and the Dedekind different of the homogeneous coordinate ring of $${\mathbb X}$$ or its Artinian reduction. We also characterize arithmetically Gorenstein schemes in novel ways and bring in further tools such as the module of regular differential forms, the fundamental class, and the Jacobian module of $${\mathbb X}$$ . Throughout we strive to work over an arbitrary base field K and keep the scheme $${\mathbb X}$$ as general as possible, thereby improving several known characterizations. [ABSTRACT FROM AUTHOR]

Let m , n ≥ 3 , ( m − 1 ) ( n − 1 ) + 2 ≤ p ≤ m n , and u = m n − p . The set R u × n × m of all real tensors with size u × n × m is one to one corresponding to the set of bilinear maps R m × R n → R u . We show that R m × n × p has plural typical ranks p and p + 1 if and only if there exists a nonsingular bilinear map R m × R n → R u . We show that there is a dense open subset O of R u × n × m such that for any Y ∈ O , the ideal of maximal minors of a matrix defined by Y in a certain way is a prime ideal and the real radical of that is the irrelevant maximal ideal if that is not a real prime ideal. Further, we show that there is a dense open subset T of R n × p × m and continuous surjective open maps ν : O → R u × p and σ : T → R u × p , where R u × p is the set of u × p matrices with entries in R , such that if ν ( Y ) = σ ( T ) , then rank T = p if and only if the ideal of maximal minors of the matrix defined by Y is a real prime ideal. [ABSTRACT FROM AUTHOR]

Letm = (m0,m1,m2,n) be an almost arithmetic sequence, i.e., a sequence of positive integers with gcd(m0,m1,m2,n) = 1, such thatm0 < m1 < m2form an arithmetic progression,nis arbitrary and they minimally generate the numerical semigroup Γ =m0ℕ +m1ℕ +m2ℕ +nℕ. Letkbe a field. The homogeneous coordinate ringk[Γ] of the affine monomial curve parametrically defined byX0 = tm0,X1 = tm1,X2 = tm2,Y = tnis a gradedR-module, whereRis the polynomial ringk[X0,X1,X2,Y] with the gradingdegXi: = mi,degY: = n. In this paper, we construct a minimal graded free resolution fork[Γ]. [ABSTRACT FROM PUBLISHER]

Multi-view Geometry is reviewed from an Algebraic Geometry perspective and multi-focal tensors are constructed as equivariant projections of the Grassmannian. A connection to the principal minor assignment problem is made by considering several flatlander cameras. The ideal of the quadrifocal variety is computed up to degree 8 (and partially in degree 9) using the representations of GL ( 3 ) × 4 in the polynomial ring on the space of 3 × 3 × 3 × 3 tensors. Further representation-theoretic analysis gives a lower bound for the number of minimal generators. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study the notion of linkage by perfect modules, which we call perfect linkage, for Cohen–Macaulay modules over Cohen–Macaulay local rings. We explore perfect linkage in connection with syzygies, maximal Cohen–Macaulay approximations and Yoshino–Isogawa linkage. We recover a theorem of Yoshino and Isogawa, and analyze the structure of double perfect linkage. Moreover, we establish a criterion for two Cohen–Macaulay modules of codimension one to be perfectly linked, and apply it to the classical linkage theory for ideals. We also construct various examples of linkage of modules and ideals. [ABSTRACT FROM AUTHOR]

We give an explicit formula for the Hilbert Series of an algebra defined by a linearly presented, standard graded, residual intersection of a grade three Gorenstein ideal. [ABSTRACT FROM PUBLISHER]

We compute the primary decomposition of certain ideals generated by subsets of minors in a generic matrix or in a generic symmetric matrix, or subsets of Pfaffians in a generic skew-symmetric matrix. Specifically, the ideals we consider are generated by minors that have at least some given number of rows and columns in certain submatrices. [ABSTRACT FROM PUBLISHER]

We present a closed formula and a simple algorithmic procedure to compute the projective dimension of square-free monomial ideals associated to string or cycle hypergraphs. As an application, among these ideals we characterize all the Cohen–Macaulay ones. [ABSTRACT FROM PUBLISHER]

Kang, Cho, and Ko gave a structure theorem for some classes of perfect ideals of grade 3 which are linked to almost complete intersections by a regular sequence. We define a 5 × 9 matrixfdetermined by three matricesA,T, andYand construct a class of perfect ideals 𝒦4(f) of grade 3 defined byf. This contains a class of perfect ideals of grade 3 minimally generated by five elements which are not Gorenstein. We give a structure theorem for some classes of perfect ideals of grade 3 linked to 𝒦4(f) by a regular sequence in 𝒦4(f). [ABSTRACT FROM PUBLISHER]

Recently, Seyed Fakhari proved that if I is a weakly polymatroidal monomial ideal in $${S\,=\,\mathbb{K}[x_1,\ldots,x_n]}$$ , then Stanley's conjecture holds for S/ I, namely, sdepth $${(S/I)\,\geq\, {\rm depth}(S/I)}$$ . We generalize his ideas and introduce several new classes of monomial ideals which also share this property. In particular, if I is the Stanley-Reisner ideal of the Alexander dual of a nonpure vertex decomposable simplicial complex, then Stanley's conjecture holds for S/ I. [ABSTRACT FROM AUTHOR]

Let k be an arbitrary field. In this note, we show that if a sequence of relatively prime positive integers a = ( a, a, a, a) defines a Gorenstein non complete intersection monomial curve $$\mathcal{C}(a)$$ in $$\mathbb{A}_k^4$$, then there exist two vectors u and v such that $$\mathcal{C}(a + tu)$$ and $$\mathcal{C}(a + tv)$$ are also Gorenstein non complete intersection affine monomial curves for almost all t ≥ 0. [ABSTRACT FROM AUTHOR]

We prove that a general polynomial form of degreedin 4 variables, over the complex field, can be written as the sum oftwodeterminants of 2 × 2 matrices of forms, with given degree matrix (aij), for any choice of non-negative integersaij ≤ dwitha11 + a22 = a12 + a21 = d. [ABSTRACT FROM AUTHOR]

Consider a complete intersectionIof type (d1,…,dr) in a polynomial ring over a field of characteristic 0. We study the graded system of ideals {gin(In)}nobtained by taking the reverse lexicographic generic initial ideals of the powers ofIand describe its asymptotic behavior. This behavior is nicely captured by thelimiting shapewhich is shown to depend only on the type of the complete intersection. [ABSTRACT FROM AUTHOR]

With a simple graph G on [n], we associate a binomial ideal PGgenerated by diagonal minors of an n × n matrix X = (xij) of variables. We show that for any graph G, PGis a prime complete intersection ideal and determine the divisor class group of K[X]/PG. By using these ideals, one may find a normal domain with free divisor class group of any given rank. [ABSTRACT FROM AUTHOR]

Wc compute the mixed multiplicities of the Rees algebra of the ideal generated by the minors of the matrix ... For that we have to solve the problem of counting lattice paths in a right triangle, which meet the diagonal a fixed number of times. [ABSTRACT FROM AUTHOR]

Buchsbaum and Eisenbud [3] proved a structure theorem for Gorenstein ideals of grade 3 which claims that every Gorenstein ideal of grade 3 is generated by the submaximal order pfaffians of an alternating matrix. In this article, we characterize some classes of grade 3 perfect ideals which are linked to an almost complete intersection of even type. Furthermore, these structure theorems give us relations between the Gorenstein sequences and the Hilbert functions of the grade 3 perfect ideals that we characterized. [ABSTRACT FROM PUBLISHER]

Let F1,..., Fn be homogeneous polynomials of positive degrees in the polynomial algebra A[T] = A[T0,..., Tn] graded by arbitrary positive integral weights for the indeterminates such that D: = A[T]/(F1,..., Fn) is a complete intersection of relative dimension 1 over the commutative noetherian ring A. Then Proj D is an affine scheme over A and its algebra B: = Γ(Proj D) of global sections is finite and stably free. A formula for the discriminant dB|A = Discr(F1,..., Fn) of B over A is given generalizing the well-known formula for the discriminant of an A-algebra of type A[X]/(G), where G is a monic polynomial in one variable. The formula is a special case of a result on discriminants for A-bilinear forms on B derived from linear forms B → A. The general formula uses the description of the linear forms on B with the help of a duality theory for D and the theory of resultants. [ABSTRACT FROM AUTHOR]

Let E and G be free modules of rank e and g, respectively, over a commutative noetherian ring R. The identity map on E* ⊗ G induces the Koszul complex ...→ SmE* ⊗ SnG ⊗ [This symbol cannot be presented in ASCII format] P (E* ⊗ G) → Sm+1E* ⊗ Sn+1 G ⊗ [This symbol cannot be presented in ASCII format] P-1 (E* ⊗ G) → ... and its dual ... → Dm+1 E ⊗ D n+1 G* ⊗ [This symbol cannot be presented in ASCII format]P-1 (E⊗G*) → DmE ⊗ DnG* ⊗ [This symbol cannot be presented in ASCII format] (E⊗G*) → ... Let HN(m,n,p) and HM(m,n,p) be the homology of the above complexes at SmE* ⊗ SnG ⊗ [This symbol cannot be presented in ASCII format] P (E* ⊗ G) and DmE ⊗ Dn G* ⊗ [This symbol cannot be presented in ASCII format] P (E ⊗G*), respectively. In this paper, we investigate the modules HN(m,n,p) and HM(m,n,p) when -e ≤ m - n ≤ g. We record the fact, already implicitly calculated by Bruns and Guerrieri, that HN(m,n,p) ≅ H M (m′,n′,p′), provided m + m′ = g - 1, n + n′ = e-1, p+p′ = (e-1)(g-1), and 1-e ≤ m - n ≤ g - 1. If m - n is equal to either g or -e, then we prove that the only nonzero modules of the form H N(m,n,p) and HM(m,n,p) appear in one of the split exact sequences. 0 → H M(g,0,p′) → [This symbol cannot be presented in ASCII format] g+p′ (E ⊗ G*) → H N(0,e,p) → 0, or 0 → H M(0,e,p′) → [This symbol cannot be presented in ASCII format] e+p′ (E ⊗ G*) → H N(g,0,p) → 0, where p + p′ = (e - 1)(g - 1) -1. The modules that we study are not always free modules. Indeed, if m=n, then the module H N(m,n,p) is equal to a homogeneous summand of the graded module Tor[This symbol cannot be presented in ASCII format](T,R), where P is a polynomial ring in eg variables over R and T is the determinantal ring defined by the 2 x 2 minors of the corresponding e x g matrix of indeterminates. Hashimoto's work shows that if e and g are both at least five, then H N(2,2,3) is not a free module when R is Z, and when R is a field, the rank of this module depends on the characteristic of R. When the modules H M (m,n,p) are free, they are summands of the resolution of the universal ring for finite length modules of projective dimension two. [ABSTRACT FROM AUTHOR]