Related papers: The Segre Determinant

A tensor product surface $\mathscr{S}$ is an algebraic surface that is defined as the closure of the image of a rational map $\phi$ from $\mathbb{P}^1\times \mathbb{P}^1$ to $\mathbb{P}^3$. We provide new determinantal representations of…

We study an irreducible real-analytic germ of an $n$-dimensional variety in $n$ dimensional complex space. Assuming that the variety is Segre nondegenerate we define an averaging operator that generalizes the Moser--Webster involution. This…

We evaluate the determinant $\det_{1\leq i,j\leq n}(\binom{x+y+j}{x-i+2j}-\binom{x+y+j}{x+i+2j})$, which gives the number of lozenge tilings of a hexagon with cut off corners. A particularly interesting feature of this evaluation is that it…

We prove the rationality of a $\k$-form $X$ of the product $S$ of projective spaces provided the existence of a $\k$-point on $X$. The method of the proof is to find a Galois-invariant birational projection of $S$ to the projective space.…

The Severi variety parameterizes plane curves of degree d with delta nodes. Its degree is called the Severi degree. For large enough d, the Severi degrees coincide with the Gromov-Witten invariants of P^2. Fomin and Mikhalkin (2009) proved…

The discriminant of a multivariate polynomial with indeterminate coefficients is not necessarily a hypersurface, and characterizing its codimension was an open problem for quite a while. We resolve this problem for the discriminants of…

Segre classes encode essential intersection-theoretic information concerning vector bundles and embeddings of schemes. In this paper we survey a range of applications of Segre classes to the definition and study of invariants of singular…

We study the problem of characterizing linear preserver subgroups of algebraic varieties, with a particular emphasis on secant varieties and other varieties of tensors. We introduce a number of techniques built on different geometric…

We propose a notion of a ternary skew-symmetric covariant tensor of 3rd order, consider it as a 3-dimensional matrix and study a ten-dimensional complex space of these tensors. We split this space into a direct sum of two five-dimensional…

In this paper, we consider a six parameter family of affine Segre surfaces embedded in $\mathbb C^6$. For generic values of the parameters, this family is associated to the $q$-difference sixth Painlev\'e equation. We show that different…

Moduli spaces of stable sheaves on smooth projective surfaces are in general singular. Nonetheless, they carry a virtual class, which -- in analogy with the classical case of Hilbert schemes of points -- can be used to define intersection…

The pseudodeterminant $\textrm{pdet}(M)$ of a square matrix is the last nonzero coefficient in its characteristic polynomial; for a nonsingular matrix, this is just the determinant. If $\partial$ is a symmetric or skew-symmetric matrix then…

The choice of a homogeneous ideal in a polynomial ring defines a closed subscheme $Z$ in a projective space as well as an infinite sequence of cones over $Z$ in progressively higher dimension projective spaces. Recent work of Aluffi…

Let $(p_n)_n$ be a sequence of orthogonal polynomials with respect to the measure $\mu$. Let $T$ be a linear operator acting in the linear space of polynomials $\PP$ and satisfying that $\dgr(T(p))=\dgr(p)-1$, for all polynomial $p$. We…

The main result of the paper is a determinantal formula for the restriction to a torus fixed point of the equivariant class of a Schubert subvariety in the torus equivariant integral cohomology ring of the Grassmannian. As a corollary, we…

We consider the scheme $X_{r,d,n}$ parametrizing $n$ ordered points in projective space $\mathbb{P}^r$ that lie on a common hypersurface of degree $d$. We show that this scheme has a determinantal structure and we prove that it is…

We study projective schemes arising from eigenvectors of tensors, called eigenschemes. After some general results, we give a birational description of the variety parametrizing eigenschemes of general ternary symmetric tensors and we…

The projective space of symmetric tensors of degree d can be reinterpreted as a projective space of finite, graded Gorenstein rings with socle in degree d. Via a pair of explicit stability conditions (one for even values of d and one for…

A theta curve is a spatial embedding of the $\theta$-graph in the three-sphere, taken up to ambient isotopy. We define the determinant of a theta curve as an integer-valued invariant arising from the first homology of its Klein cover. When…

Let E and G be free modules of rank e and g, respectively, over a commutative noetherian ring R. The identity map on E^* tensor G induces the Koszul complex ... -> S_mE^* tensor S_nG tensor Wedge^p(E^* tensor G) -> S_{m+1}E^* tensor…