Combinatorial symbolic powers

Symbolic powers are studied in the combinatorial context of monomial ideals. When the ideals are generated by quadratic squarefree monomials, the generators of the symbolic powers are obstructions to vertex covering in the associated graph and its blowups. As a result, perfect graphs play an important role in the theory, dual to the role played by perfect graphs in the theory of secants of monomial ideals. We use Gröbner degenerations as a tool to reduce questions about symbolic powers of arbitrary ideals to the monomial case. Among the applications are a new, unified approach to the Gröbner bases of symbolic powers of determinantal and Pfaffian ideals.     

Ordinary and symbolic powers are Golod

Let S$S$ be a positively graded polynomial ring over a field of characteristic 0$0$, and I⊂S$I\subset S$ a proper graded ideal. In this note it is shown that S/I$S/I$ is Golod if ∂(I)²⊂I$\partial {\left(I\right)}^2\subset I$. Here ∂(I)$\partial \left(I\right)$ denotes the ideal generated by all the partial derivatives of elements of I$I$. We apply this result to find large classes of Golod ideals, including powers, symbolic powers, and saturations of ideals.     

A note on symbolic and ordinary powers of homogeneous ideals

In this note we are interested in the graded modulesM _k=I^(k)/I^k and , whereI is a saturated ideal in the homogeneous coordinate ringS=K[x₀,…,x_n] of ℙⁿ,I ^(k) is the symbolic power and is the saturation of the ordinary power. Very little is known about these modules, and we provide a bound on their diameters, we compute the Hilbert functions and we study some characteristic submodules in the special case ofn+1 general points in ℙⁿ.

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Sunto In questa nota siamo interessati ai moduli graduatiM _k=I(k)/I^k e , doveI è un ideale saturato nell'anello delle coordinate omogeneeS:=K[x₀,…,x_n] di ℙⁿ,I ^(k) è la potenza simbolica e è la saturazione della potenza ordinaria. Poco è noto su questi moduli e qui viene fornito un limite superiore ai loro diametri. Ne calcoliamo inoltre le funzioni di Hilbert e studiamo alcuni sottomoduli caratteristici nel caso speciale din+1 punti in posizione generale, in ℙⁿ.

     

Links of symbolic powers of prime ideals

In this paper, we prove the following. Let be a Cohen-Macaulay local ring of dimension d ≥ 2. Suppose that either R is not regular or if R is regular assume that d ≥ 3. Let t ≥ 2 be a positive integer. If is a regular sequence contained in , then

Geometric nullstellensatz and symbolic powers on arbitrary varieties

In recent years, a multiplier ideal defined on arbitrary varieties, so called Mather–Jacobian multiplier ideal, has been developed independently by Ein–Ishii–Mustata and de Fernex–Docampo. With this new tool, we have a chance of extending some classical results proved in nonsingular case to arbitrary varieties to establish their general forms. In this paper, we first extend a result of geometric nullstellensatz due to Ein–Lazarsfeld in nonsingular case to any projective varieties. Then we prove a result on comparison of symbolic powers with ordinary powers on any varieties, which extends results of Ein–Lazarsfeld–Smith and Hochster–Huneke.     

Fuzzy sets: A topos-logical point of view

Bénabou deduction-categories are defined, with a set of additional assumptions that define categories with formal finite limits (resp. formal regular categories, formal logoi, formal topoi). They are shown to be generalized structures in which higher-order many-sorted languages can be realized. The corresponding Gentzen-type higher-order calculus of sequents is explicited and the soundness theorem is formulated. A construction is given, which associates to each deduction category with formal properties a real category with the corresponding real properties, in a universal way. The corresponding sounddess and completeness properties are formulated for the real categories thus obtained. Fuzzy sets, as generalized by Goguen are introduced, considered as the objects of a category Fuz(H), which turns out to be the real category associated to a very simple formal topos, and thus to be itself a topos: furthermore this is proved to be a Grothendieck topos which is a strictly full epireflective subcategory of Higgs' category of ‘H-valued sets’. Topoi are proposed as generalized fuzzy sets, and deductio0-categories as generalized² fuzzy sets. Some related topics such as Arbib-Manes fuzzy theories, probability, many-valued and fuzzy logics, intensional logic are very briefly touched upon.     

Linear subspaces,symbolic powers and Nagata type conjectures

Inspired by results of Guardo, Van Tuyl and the second author for lines in P³${\mathbb{P}}^3$, we develop asymptotic upper bounds for the least degree of a homogeneous form vanishing to order at least m on a union of disjoint r  -dimensional planes in Pⁿ${\mathbb{P}}^n$ for n?2r+1$n?2r+1$. These considerations lead to new conjectures that suggest that the well known conjecture of Nagata for points in P²${\mathbb{P}}^2$ is not an exotic statement but rather a manifestation of a much more general phenomenon which seems to have been overlooked so far.     

Comparing symbolic powers of edge ideals of weighted oriented graphs

Journal of Algebraic Combinatorics - Let D be a weighted oriented graph and I(D) be its edge ideal. If D contains an induced odd cycle of length $$2n+1$$ , under certain condition, we show that $$...     

The cleanness of (symbolic) powers of Stanley-Reisner ideals

Let Δ be a pure simplicial complex on the vertex set [n] = {1,..., n} and I _Δ its Stanley-Reisner ideal in the polynomial ring S = K[x ₁,..., x _n]. We show that Δ is a matroid (complete intersection) if and only if S/I _Δ ^(m) (S/I _Δ ^(m)) is clean for all m ∈ N and this is equivalent to saying that S/I _Δ ^(m) (S/I _Δ ^(m), respectively) is Cohen-Macaulay for all m ∈ N. By this result, we show that there exists a monomial ideal I with (pretty) cleanness property while S/I ^m or S/I ^m is not (pretty) clean for all integer m ≥ 3. If dim(Δ) = 1, we also prove that S/I _Δ ⁽²⁾ Δ (S/I _Δ ²) is clean if and only if S/I _Δ ⁽²⁾ (S/I _Δ ², respectively) is Cohen-Macaulay.     

The topology determined by the symbolic powers of primary ideals

Let q be a p-primary ideal in a Noetherian ring R. The main theorem characterizes when the q-adic and q-symbolic topologies on R are linearly equivalent; that is, when there n kexists an integer k≥0 such that q⁽ⁿ⁾ ?q^n-k for all n≥k. Using this, it is shown that when this holds for q, then it holds for several other primary ideals related to q (both in R and in certain other rings related to R) and that the powers of q(ⁿ) are symbolic powers for all n≥k (so p has primary ideals all of whose powers are symbolic powers).     

The Buchsbaum property of symbolic powers of Stanley-Reisner ideals of dimension 1

Let S be a polynomial ring and I be the Stanley-Reisner ideal of a simplicial complex Δ. The purpose of this paper is investigating the Buchsbaum property of S/I^(r) when Δ is pure dimension 1. We shall characterize the Buchsbaumness of S/I^(r) in terms of the graphical property of Δ. That is closely related to the characterization of the Cohen-Macaulayness of S/I^(r) due to the first author and N.V. Trung.