Matrix Of Linear Forms Research Articles

EXAMPLES OF SMOOTH SURFACES IN ℙ3 WHICH ARE ULRICH-WILD

Let <TEX>$F{\subseteq}{\mathbb{P}}^3$</TEX> be a smooth surface of degree <TEX>$3{\leq}d{\leq}9$</TEX> whose equation can be expressed as either the determinant of a <TEX>$d{\times}d$</TEX> matrix of linear forms, or the pfaffian of a <TEX>$(2d){\times}(2d)$</TEX> matrix of linear forms. In this paper we show that F supports families of dimension p of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrary large p.

A matrix of linear forms which is annihilated by a vector of indeterminates

Let R=k[T1,…,Tf] be a standard graded polynomial ring over the field k and Ψ be an f×g matrix of linear forms from R, where 1≤g<f. Assume [T1⋯Tf]Ψ is 0 and that gradeIg(Ψ) is exactly one short of the maximum possible grade. We resolve R‾=R/Ig(Ψ), prove that R‾ has a g-linear resolution, record explicit formulas for the h-vector and multiplicity of R‾, and prove that if f−g is even, then the ideal Ig(Ψ) is unmixed. Furthermore, if f−g is odd, then we identify an explicit generating set for the unmixed part, Ig(Ψ)unm, of Ig(Ψ), resolve R/Ig(Ψ)unm, and record explicit formulas for the h-vector of R/Ig(Ψ)unm. (The rings R/Ig(Ψ) and R/Ig(Ψ)unm automatically have the same multiplicity.) These results have applications to the study of the blow-up algebras associated to linearly presented grade three Gorenstein ideals.

Homological projective duality for determinantal varieties

In this paper we prove Homological Projective Duality for categorical resolutions of several classes of linear determinantal varieties. By this we mean varieties that are cut out by the minors of a given rank of a m×n matrix of linear forms on a given projective space. As applications, we obtain pairs of derived-equivalent Calabi–Yau manifolds, and address a question by A. Bondal asking whether the derived category of any smooth projective variety can be fully faithfully embedded in the derived category of a smooth Fano variety. Moreover we discuss the relation between rationality and categorical representability in codimension two for determinantal varieties.

A Lower Bound for the Determinantal Complexity of a Hypersurface

We prove that the determinantal complexity of a hypersurface of degree $$d > 2$$d>2 is bounded below by one more than the codimension of the singular locus, provided that this codimension is at least 5. As a result, we obtain that the determinantal complexity of the $$3 \times 3$$3×3 permanent is 7. We also prove that for $$n> 3$$n>3, there is no nonsingular hypersurface in $${\mathbb {P}}^n$$Pn of degree d that has an expression as a determinant of a $$d \times d$$d×d matrix of linear forms, while on the other hand for $$n \le 3$$n≤3, a general determinantal expression is nonsingular. Finally, we answer a question of Ressayre by showing that the determinantal complexity of the unique (singular) cubic surface containing a single line is 5.

Determinantal representations of hyperbolic plane curves: An elementary approach

In 2007, Helton and Vinnikov proved that every hyperbolic plane curve has a definite real symmetric determinantal representation. By allowing for Hermitian matrices instead, we are able to give a new proof that relies only on the basic intersection theory of plane curves. We show that a matrix of linear forms is definite if and only if its co-maximal minors interlace its determinant and extend a classical construction of determinantal representations of Dixon from 1902. Like the Helton–Vinnikov theorem, this implies that every hyperbolic region in the plane is defined by a linear matrix inequality.

Aluffi torsion-free ideals

A special class of algebras which are intermediate between the symmetric and the Rees algebras of an ideal was introduced by P. Aluffi in 2004 to define characteristic cycle of a hypersurface parallel to conormal cycle in intersection theory. These algebras are recently investigated by A. Nasrollah Nejad and A. Simis who named them Aluffi algebras. For a pair of ideals J ⊆ I of a commutative ring R, the Aluffi algebra of I / J is called Aluffi torsion-free if it is isomorphic to the Rees algebra of I / J . In this paper, ideals generated by 2-minors of a 2 × n matrix of linear forms and also edge ideals of graphs are considered and some conditions are presented which are equivalent to Aluffi torsion-free property of them.

On the untwisting of general de Jonquières and cubo-cubic Cremona transformations of $${\mathbb{P}^3}$$

We consider Cremona transformations of \({\mathbb{P}^3}\) which either stabilize the set of planes passing through a point or they are defined by the maximal minors of a 4 × 3 matrix of linear forms. For a general transformation satisfying one of these two conditions we give an explicit decomposition as a product of elementary links; in other words, we perform the so-called Sarkisov program for (general) transformations belonging to these two classes. Conversely, we show that a Cremona transformation whose decomposition coincides with one of those we have obtained belongs to one of these classes.

On varieties of almost minimal degree II: A rank-depth formula

Let $X \subset \mathbb P^r_K$ denote a variety of almost minimal degree other than a normal del Pezzo variety. Then $X$ is the projection of a rational normal scroll $\tilde X \subset {\mathbb P}^{r+1}_K$ from a point $p \in {\mathbb P}^{r+1}_K \setminus \tilde X.$ We show that the arithmetic depth of $X$ can be expressed in terms of the rank of the matrix $M’(p),$ where $M’$ is the matrix of linear forms whose $3\times 3$ minors define the secant variety of $\tilde X.$

Implicitization and parametrization of nonsingular cubic surfaces

In this paper we unify the two subjects of implicitization and parametrization of nonsingular cubic surfaces. Beginning with a cubic parametrization with six basepoints, we first form a three by four Hilbert–Burch matrix, and then a three by three matrix of linear forms whose determinant is the implicit equation. Beginning with an implicit equation, we show how to construct a three by three matrix of linear forms whose determinant is the implicit equation, and from it construct the Hilbert–Burch matrix and a parametrization. The intermediate three by three matrix is shown to contain information about lines and cubic curves that lie on the surface, as well as to aid in the construction of inversion formulas.

Gröbner basis and free resolution of the ideal of 2-minors of a 2 × n matrix of linear forms*

We give a Gröbner basis for the ideal of 2-minors of a 2 × n utiatrix of linear forms. The minimal free resolution of such an ideal is obtained in [4] when the corresponding Kronecker-Weierstrass normal form has no iiilpotent blocks. For the general case, using this result, the Grobner basis and the Eliahou-Kervaire resolution for stable monomial ideals, we obtain a free resolution with the expected regularity. For a specialization of the defining ideal of ordinary pinch points, as a special case of these ideals, we provide a minimal free resolution explicitly in terms of certain Koszul complex.

Problème de Brill-Noether pour les fibrés de steiner. Applications aux courbes gauches

A Steiner bundle over the projective 3-space is the kernel in a trivial bundle of a morphism defined by a matrix of linear forms. We produce various Steiner bundles E of rank n such that E(1) has n - 1 sections, the dependency locus of which is a smooth curve.

Parametrization of the orbits of cubic surfaces

The aim of the paper is the study of the orbits of the action of PGL4 on the space ℙ3 of the cubic surfaces of ℙ3, i.e., the classification of cubic surfaces up to projective motions. A varietyQ⊂ℙ19 is explicitely constructed as the union of 22 disjoint irreducible components which are either points or open subsets of linear spaces. More precisely, each orbit of the above action intersects one componentX ofQ in a finite number of points and the action of PGL4 restricted on each componentX is equivalent to the action of a finite groupG X onX which can be explicitely computed. Finally the cubic surfaces of each component ofQ are studied in details by determining their stabilizers, their rational representations and whether they can be expressed as the determinant of a 3×3 matrix of linear forms. The results are obtained with computational techniques and with the aid of some computer algebra systems like CoCoA, Macaulay and Maple.

Self-polar double configurations in projective geometry: I. A general condition for self-polarity

Let be a p×q matrix of linear forms in the n+1 coordinates in a projective space Πn. Then points which satisfy the q equations in general span a space Πn-q, but will span a space Πn-q+1 if a set μ,={μβ of multipliers can be found such that Such a set μ can be found if and only if the equations have solutions.