Let X be a smooth hypersurface in projective space. We discuss in this paper when X can be defined by an equation det M = 0 (resp. pf M = 0), where M is a matrix (resp. a skew-symmetric matrix) with homogeneous entries. Standard homological algebra methods show that this is equivalent to produce a line bundle (resp. a rank 2 vector bundle) E of a certain type on X . We discuss a number of applications for hypersurfaces of small dimension. An Appendix by F.-O. Schreyer proves (using Macaulay 2) that a general form of degree d in P^3 (resp. P^4) can be written as the pfaffian of a skew-symmetric (2d)x(2d) matrix with linear entries in the expected range, that is d < 16 (resp. d < 6).
We prove that a general determinantal hypersurface of dimension 3 is nodal. Moreover, in terms of Chern classes associated with bundle morphisms, we derive a formula for the intersection homology Euler characteristic of a general determinantal hypersurface.
A (global) determinantal representation of hypersurface in P^n is a matrix, whose entries are linear forms in homogeneous coordinates and whose determinant defines the hypersurface. We study the properties of such representations for singular (possibly reducible or non-reduced) hypersurfaces. In particular, we obtain the decomposability criteria for determinantal representations of globally reducible hypersurfaces. Further, we classify the determinantal representations in terms of the corresponding kernel sheaves on $X$. Finally, we extend the results to the case of symmetric/self-adjoint repr…
In the present note we study determinantal arrangements constructed with use of the $3$-minors of a $3 \times 5$ generic matrix of indeterminates. In particular, we show that certain naturally constructed hypersurface arrangements in $\mathbb{P}^{14}_{\mathbb{C}}$ are free.
We study the birational geometry of irreducible holomorphic symplectic varieties arising as varieties of lines of general cubic fourfolds containing a cubic scroll. We compute the ample and moving cones, and exhibit a birational automorphism of infinite order explaining the chamber decomposition of the moving cone.
The characteristic polynomial of an $r$-tuple $(A_1,..., A_r)$ of $n \times n$ matrices is the determinant $\det(x_0 I + x_1 A_1 + ... + x_r A_r)$. We show that if $r$ is at least 3 and $A = (A_1,..., A_r)$ is an $r$-tuple of matrices in general position, then up to conjugacy there are only finitely many $r$-tuples of matrices with the same characteristic polynomial as $A$. Equivalently, the locus of determinantal hypersurfaces of degree $n$ in $P^r$ is irreducible of dimension $(r-1)n^2 + 1$.
We prove that the determinantal complexity of a hypersurface of degree $d > 2$ 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 $3 \times 3$ permanent is $7$. We also prove that for $n> 3$, there is no nonsingular hypersurface in $\mathbf{P}^n$ of degree $d$ that has an expression as a determinant of a $d \times d$ matrix of linear forms while on the other hand for $n \le 3$, a general determinantal expression is nonsingular. Finally, we answer a question of R…
A hypersurface $H$ in $\mathbb{P}^r$ of degree $n$ is called determinantal if it is the zero locus of a polynomial of the form $\operatorname{det}(x_0 A_0 + \ldots + x_r A_r)$ for some $(r+1)$-tuple of $n \times n$ matrices $A = (A_0, \ldots, A_r)$. We will refer to $A$ as a presentation of $H$. Another presentation $B = (B_0, B_1, \ldots, B_r)$ of $H$ can be obtained by choosing $g_1, g_2 \in \operatorname{GL}_n$ and setting $B_i = g_1 A_i g_2$ for every $i = 0, 1, \ldots, r$. In this case $A$ and $B$ are called equivalent. The second author and A. Vistoli have shown that for $r \geq 3$ a gen…
Let M be a matrix whose entries are power series in several variables and determinant det(M) does not vanish identically. The equation det(M)=0 defines a hypersurface singularity and the (co)-kernel of M is a maximally Cohen-Macaulay module over the local ring of this singularity. Suppose the determinant det(M) is reducible, i.e. the hypersurface is locally reducible. A natural question is whether the matrix is equivalent to a block-diagonal or at least to an upper-block-triangular. (Or whether the corresponding module is decomposable or at least is an extension.) We give various necessary and…
From a block-diagonal $(n+1) \times (m+1) \times (m+1)$ tensor symmetric in the last two entries one obtains two varieties: an intersection of symmetric determinantal hypersurfaces $X$ in $n$-dimensional projective space, and an intersection of quadrics $\mathfrak{C}$ in $m$-dimensional projective space. Under mild technical assumptions, we characterize the accidental singularities of $X$ in terms of $\mathfrak{C}$. We apply our methods to algebraic curves and show how to construct theta characteristics of certain canonical curves of genera 3, 4, and 5, generalizing a classical construction of…
Given a finite generating set $T=\{g_0,\dots, g_n\}$ of a group $G$, and a representation $ρ$ of $G$ on a Hilbert space $V$, we investigate how the geometry of the set $D(T,ρ)=\{ [x_0 : \dots : x_n] \in\mathbb C\mathbb P^n \mid \sum x_iρ(g_i) \text{ not invertible} \}$ reflects the properties of $ρ$. When $V$ is finite-dimensional this is an algebraic hypersurface in $\mathbb C\mathbb P^n$. In the special case $T=G$ and $ρ=$ the left regular representation of $G$, this hypersurface is defined by the \emph{group determinant}, an object studied extensively in the founding work of Frobenius that …
If a real symmetric matrix of linear forms is positive definite at some point, then its determinant is a hyperbolic hypersurface. In 2007, Helton and Vinnikov proved a converse in three variables, namely that every hyperbolic plane curve has a definite real symmetric determinantal representation. The goal of this paper is to give a more concrete proof of a slightly weaker statement. Here we show that every hyperbolic plane curve has a definite determinantal representation with Hermitian matrices. We do this by relating the definiteness of a matrix to the real topology of its minors and extendi…
Let $X={\bf P}^2\times{\bf P}^{n-1}$ embedded with $Ø(1,2)$. We prove that its $(n+1)$-secant variety $σ_{n+1}(X)$ is a hypersurface, while it is expected that it fills the ambient space. The equation of $σ_{n+1}(X)$ is the symmetric analog of the Strassen equation. When $n=4$ the determinantal map takes $σ_5(X)$ to the hypersurface of Lüroth quartics, which is the image of the Barth map studied by LePotier and Tikhomirov. This hint allows to obtain some results on the jumping lines and the Brill-Noether loci of symplectic bundles on ${\bf P}^2$ by using the higher secant varieties of $X$.
In this note we study standard and good determinantal schemes. We show that there exist arithmetically Cohen-Macaulay schemes that are not standard determinantal, and whose general hyperplane section is good determinantal. We prove that if a general hyperplane section of a scheme is standard (resp. good) determinantal, then the scheme is standard (resp. good) determinantal up to flat deformation. We also study the transfer of the property of being standard or good determinantal under basic double links.
One studies certain degenerations of the generic square matrix over a field $k$ along with its main related structures, such as the determinant of the matrix, the ideal generated by its partial derivatives, the polar map defined by these derivatives, the Hessian matrix and the ideal of the submaximal minors of the matrix. The main tool comes from commutative algebra, with emphasis on ideal theory and syzygy theory. The structure of the polar map is completely identified and the main properties of the ideal of submaximal minors are determined. Cases where the degenerated determinant has non-van…
A scheme $X\subset \PP^{n+c}$ of codimension $c$ is called {\em standard determinantal} if its homogeneous saturated ideal can be generated by the maximal minors of a homogeneous $t \times (t+c-1)$ matrix and $X$ is said to be {\em good determinantal} if it is standard determinantal and a generic complete intersection. Given integers $a_0,a_1,...,a_{t+c-2}$ and $b_1,...,b_t$ we denote by $W(\underline{b};\underline{a})\subset \Hi ^p(\PP^{n+c})$ (resp. $W_s(\underline{b};\underline{a})$) the locus of good (resp. standard) determinantal schemes $X\subset \PP^{n+c}$ of codimension $c$ defined by …
We show that on a certain hypersurface in P3 there is a (q3 + q2 + q + 1)q+1-symmetric configuration (resp. a ((q3 + 1)(q2 + 1)q+1, (q3 + 1)(q + 1)q2+1)) -configuration) made up of the rational points over Fq (resp. over Fq2) and the lines over Fq (resp. the lines over Fq2). We also determine the structure of a Lefschetz pencil made by using a line from these lines.
Let $X$ be a smooth quintic hypersurface in $\mathbb{P}^3$, let $C$ be a smooth hyperplane section of $X$, and let $H=\mathcal{O}_X(C)$. In this paper, we give a necessary and sufficient condition for the line bundle given by a non-zero effective divisor on $X$ to be initialized and aCM with respect to $H$.
We prove a determinantal formula and Pfaffian formulas that respectively describe the $K$-theoretic degeneracy loci classes for Grassmann bundles and for symplectic Grassmann and odd orthogonal bundles. The former generalizes Damon--Kempf--Laksov's determinantal formula and the latter generalize Pragacz--Kazarian's formula for the Chow ring. As an application, we introduce the factorial $GΘ/ GΘ'$-functions representing the torus equivariant $K$-theoretic Schubert classes of the symplectic and the odd orthogonal Grassmannians, which generalize the (double) theta polynomials of Buch--Kresch--Tam…
Generalizing a classical lemma of Castelnuovo, we characterize rational normal curves (resp. linearly normal elliptic curves) as curves $C\subset \PP^n$ such that the number of linearly independent hypersurfaces $Z\supset C$ of given degree~$m$ is maximal (resp. next to maximal).
