arXiv subjects
Publications and source records attributed to Stefan Tohaneanu.
The ideals generated by fold products of linear forms are generalizations of powers of defining ideals of star configurations, or of Veronese type ideals, and in this paper we study their Betti numbers. In earlier work, the authors together with Yu Xie showed that these ideals have linear graded free resolution, and in this paper we take the study of the homological information about these ideals to the next natural level.
Yuzvinsky and Rose-Terao have shown that the homological dimension of the gradient ideal of the defining polynomial of a generic hyperplane arrangement is maximum possible. In this work one provides yet another proof of this result, which in addition is totally different from the one given by Burity-Simis-Tohaneanu. Another main drive of the paper concerns a version of the above result in the case of a product of general forms of arbitrary degrees (in particular, transverse ones). Finally, some relevant cases of non general forms are also contemplated.
Let \({\mathbb K}\) be any field, let \(X\subset {\mathbb P}^{k-1}\) be a set of \(n\) distinct \({\mathbb K}\)-rational points, and let \(a\geq 1\) be an integer. In this paper we find lower bounds for the minimum distance \(d(X)_a\) of the evaluation code of order \(a\) associated to \(X\). The first results use \(α(X)\), the initial degree of the defining ideal of \(X\), and the bounds are true for any set \(X\). In another result we use \(s(X)\), the minimum socle degree, to find a lower bound for the case when \(X\) is in general linear position. In both situations we improve and generalize known results.
In this paper we give full classification of rank 3 line arrangements in $\mathbb P^2$ (over a field of characteristic 0) that have a minimal logarithmic derivation of degree 3. The classification presents their defining polynomials, up to a change of variables, with their corresponding affine pictures. We also analyze the shape of such a logarithmic derivation, towards obtaining criteria for a line arrangement to possess a cubic minimal logarithmic derivation.
Let $\mathcal{A}$ denote a central hyperplane arrangement of rank $n$ in affine space $\mathbb{K}^n$ over an infinite field $\mathbb{K}$ and let $l_1,\ldots, l_m\in R:= \mathbb K[x_1,\ldots,x_n]$ denote the linear forms defining the corresponding hyperplanes, along with the corresponding defining polynomial $f:=l_1\cdots l_m\in R$. Let $J_f$ denote the ideal generated by the partial derivatives of $f$ and let $\mathbb{I}$ designate the ideal generated by the $(m-1)$-fold products of $l_1,\ldots, l_m$. This paper is centered on the relationship between the two ideals $J_f, \mathbb{I}\subset R$, their properties and two conjectures related to them. Some parallel results are obtained in the case of forms of higher degrees provided they fulfill a certain transversality requirement.
Let $Σ$ be a finite collection of linear forms in $\mathbb K[x_0,\ldots,x_n]$, where $\mathbb K$ is a field. Denote ${\rm Supp}(Σ)$ to be the set of all nonproportional elements of $Σ$, and suppose ${\rm Supp}(Σ)$ is generic, meaning that any $n+1$ of its elements are linearly independent. Let $1\leq a\leq |Σ|$. In this article we prove the conjecture that the ideal generated by (all) $a$-fold products of linear forms of $Σ$ has linear graded free resolution. As a consequence we prove the Geramita-Harbourne-Migliore conjecture concerning the primary decomposition of ordinary powers of defining ideals of star configurations, and we also determine the resurgence of these ideals.
In this note we show that Harbourne's conjecture is true for symbolic powers of ideals of points, we check that the stable version of this conjecture is valid for ideals of very general points (resp. generic points) in $\mathbb P_{\mathbb K}^N$ (resp. $\mathbb P_{\mathbb K(\underline{z})}^N$). We also show that this conjecture and the Harbourne-Huneke conjecture are true for a class of ideals $I$ defining fat points obtained from line arrangements in $\mathbb P_{\mathbb K}^2$.
Let $\mathbb K$ be a field of characteristic 0. Given $n$ linear forms in $R=\mathbb K[x_1,\ldots,x_k]$, with no two proportional, in one of our main results we show that the ideal $I\subset R$ generated by all $(n-2)$-fold products of these linear forms has linear graded free resolution. This result helps determining a complete set of generators of the symmetric ideal of $I$. Via Sylvester forms we can analyze from a different perspective the generators of the presentation ideal of the Orlik-Terao algebra of the second order; this is the algebra generated by the reciprocals of the products of any two (distinct) of the linear forms considered. We also show that when $k=2$, and when the collection of $n$ linear forms may contain proportional linear forms, for any $1\leq a\leq n$, the ideal generated by $a$-fold products of these linear forms has linear graded free resolution.
In these notes we show that any projective subspace arrangement can be realized as a generalized star configuration variety. This type of interpolation result may be useful in designing linear codes with prescribed codewords of minimum weight, as well as in answering a couple of questions asked by the author in previous work, about the number of equations needed to define a generalized star configuration.
From the generating matrix of a linear code one can construct a sequence of generalized star configurations which are strongly connected to the generalized Hamming weights and the underlying matroid of the code. When the code is MDS, the matrix is generic and we obtain the usual star configurations. In our main result, we show that the degree of a generalized star configuration as a projective scheme is determined by the Tutte polynomial of the code. In the process, we obtain preliminary results on the primary decomposition of the defining ideals of these schemes. Additionally, we conjecture that these ideals have linear minimal free resolutions and prove partial results in this direction.
It is shown that the Orlik-Terao algebra is graded isomorphic to the special fiber of the ideal $I$ generated by the $(n-1)$-fold products of the members of a central arrangement of size $n$. This momentum is carried over to the Rees algebra (blowup) of $I$ and it is shown that this algebra is of fiber-type and Cohen-Macaulay. It follows by a result of Simis-Vasconcelos that the special fiber of $I$ is Cohen-Macaulay, thus giving another proof of a result of Proudfoot-Speyer about the Cohen-Macauleyness of the Orlik-Terao algebra.
Let $\mathcal A\subset\mathbb P^{k-1}$ be a rank $k$ arrangement of $n$ hyperplanes, with the property that any $k$ of the defining linear forms are linearly independent (i.e., $\mathcal A$ is called $k-$generic). We show that for any $j=0,\ldots,k-2$, the subspace arrangement with defining ideal generated by the $(n-j)-$fold products of the defining linear forms of $\mathcal A$ is a set-theoretic complete intersection, which is equivalent to saying that star configurations have this property.
The simple interpretation of the minimum distance of a linear code obtained by De Boer and Pellikaan, and later refined by the second author, is further developed through the study of various finitely generated graded modules. We use the methods of commutative/homological algebra to find connections between the minimum distance and the $α$-invariant of such modules.
In these notes we study hyperplane arrangements having at least one logarithmic derivation of degree two that is not a combination of degree one logarithmic derivations. It is well-known that if a hyperplane arrangement has a linear logarithmic derivation not a constant multiple of the Euler derivation, then the arrangement decomposes as the direct product of smaller arrangements. The next natural step would be to study arrangements with non-trivial quadratic logarithmic derivations. On this regard, we present a computational lemma that leads to a full classification of hyperplane arrangements of rank 3 having such a quadratic logarithmic derivation. These results come as a consequence of looking at the variety of the points dual to the hyperplanes in such special arrangements.
We show that errors in data transmitted through linear codes can be thought of as codewords of minimum weight of new linear codes. To determine errors we can then use methods specific to finding such special codewords. One of these methods consists of finding the primary decomposition of the saturation of a certain homogeneous ideal. When good words (i.e. vectors with a unique nearest neighbor) are error-corrected, the saturated ideal is just the prime ideal of a point (so the primary decomposition is superfluously determined); we show that this ideal can be computed by coloning the original homogeneous ideal with a power of a certain variable. We then determine the smallest such power for any linear code.
Given a rank 3 real arrangement $\mathcal A$ of $n$ lines in the projective plane, the Dirac-Motzkin conjecture (proved by Green and Tao in 2013) states that for $n$ sufficiently large, the number of simple intersection points of $\mathcal A$ is greater than or equal to $n/2$. With a much simpler proof we show that if $\mathcal A$ is supersolvable, then the conjecture is true for any $n$ (a small improvement of original conjecture). The Slope problem (proved by Ungar in 1982) states that $n$ non-collinear points in the real plane determine at least $n-1$ slopes; we show that this is equivalent to providing a lower bound on the multiplicity of a modular point in any (real) supersolvable arrangement. In the second part we find connections between the number of simple points of a supersolvable line arrangement, over any field of characteristic 0, and the degree of the reduced Jacobian scheme of the arrangement. Over the complex numbers even though the Sylvester-Gallai theorem fails to be true, we conjecture that the supersolvable version of the Dirac-Motzkin conjecture is true.
One studies the structure of the Rees algebra of an almost complete intersection monomial ideal of finite co-length in a polynomial ring over a field, assuming that the least pure powers of the variables contained in the ideal have the same degree. It is shown that the Rees algebra has a natural quasi-homogeneous structure and its presentation ideal is generated by explicit Sylvester forms. A consequence of these results is a proof that the Rees algebra is almost Cohen--Macaulay, thus answering affirmatively an important case of a conjecture of W. Vasconcelos.
The Orlik-Solomon algebra is the cohomology ring of the complement of a hyperplane arrangement A in C^n; it is the quotient of an exterior algebra E(V) on |A| generators. Orlik and Terao introduced a commutative analog S(V)/I of the Orlik-Solomon algebra to answer a question of Aomoto and showed the Hilbert series depends only on the intersection lattice L(A). Motivated by topological considerations, Falk and Randell introduced the property of 2-formality; we study the relation between 2-formality and the Orlik-Terao algebra. Our main result is a necessary and sufficient condition for 2-formality in terms of the quadratic component I_2 of the Orlik-Terao ideal I: 2-formality is determined by the tangent space T_p(V(I_2)) at a generic point p.