Related papers: Determinantal representation and subschemes of gen…

Here we study the space of real hyperbolic plane curves that are invariant under actions of the cyclic and dihedral groups and show they have determinantal representations that certify this invariance. We show an analogue of Nuij's theorem…

For $m \in \mathbb{N}$, we determine the irreducible components of the $m$-th Jet Scheme of a complex branch $C$ and give formulas for their number $N(m)$ and for their codimensions, in terms of $m$ and the generators of the semigroup of…

In this note we determine, for an arbitrary but a fixed degree $d$, an algorithm to list the possible values $m$ for which $M_g^{Pl}(\mathbb{Z}/m)$ is non-empty where $\mathbb{Z}/m$ denotes the cyclic group of order $m$. In particular, we…

Let $k$ be an arbitrary field and $d$ a positive integer. For each degenerate symmetric or antisymmetric bilinear form $M$ on $k^{d}$ we determine the structure of the Lie algebra of matrices that preserve $M$, and of the Lie algebra of…

The main purpose of this paper is to prove the existence of the moduli space parameterizing the embedded curve singularities of $(k^N,0)$ with an admissible Hilbert polynomial and to study its basic properties.

We study higher order determinantal varieties obtained by considering generic $m\times n$ ($m \le n$) matrices over rings of the form $F[t]/(t^k)$, and for some fixed $r$, setting the coefficients of powers of $t$ of all $r \times r$ minors…

We study $n$-dimensional matrices with $\{0,1\}$-entries ($n$-cubes) such that all their $2$-dimensional slices are incidence matrices of symmetric designs. A known construction of these objects obtained from difference sets is generalized…

We unify Linear Algebra by proposing a definition of determinants via one equation that implies all known properties of them:\\ 1. Cramer's Rule,\\ 2. Cofactor expansion,\\ 3. Antisymmetry of determinants,\\ 4. Linearity of determinants,\\…

The squared singular values of the product of $M$ complex Ginibre matrices form a biorthogonal ensemble, and thus their distribution is fully determined by a correlation kernel. The kernel permits a hard edge scaling to a form specified in…

In this paper, we introduce the notion of the diagonal property and the weak point property for an ind-variety. We prove that the ind-varieties of higher rank divisors of integral slopes on a smooth projective curve have the weak point…

Determinantal varieties are important objects of study in algebraic geometry. In this paper, we will investigate them using the jet scheme approach. We have found a new connection for the Hilbert series between a determinantal variety and…

The Zariski closure of the central path which interior point algorithms track in convex optimization problems such as linear, quadratic, and semidefinite programs is an algebraic curve. The degree of this curve has been studied in relation…

In this paper we give a new proof of the fact that for all pairs of positive integers (d, m) with d/m < 117/37, the linear system of plane curves of degree d with ten general base points of multiplicity m is empty.

We define higher order fundamental forms and osculating spaces of projective algebraic varieties, using sheaves of principal parts. We show that the $m$th fundamental form can be viewed as the differential of the $(m-1)$th Gauss map, and…

We consider the problem of writing real polynomials as determinants of symmetric linear matrix polynomials. This problem of algebraic geometry, whose roots go back to the nineteenth century, has recently received new attention from the…

Given a graded associative algebra $A$, its lower central series is defined by $L_1 = A$ and $L_{i+1} = [L_i, A]$. We consider successive quotients $N_i(A) = M_i(A) / M_{i+1}(A)$, where $M_i(A) = AL_i(A) A$. These quotients are direct sums…

In this paper, we introduce the notion of the diagonal property and the weak point property for an ind-variety. We prove that the ind-varieties of higher rank divisors of integral slopes on a smooth projective curve have the weak point…

Let $\M_g$ be the course moduli space of complex projective nonsingular curves of genus $g$. We prove that when the Brill-Noether number $\rho(g,1,n)$ is non-negative the Petri locus $P^1_{g,n}\subset \M_g$ has a divisorial component whose…

We define the generalized hypergeometric polynomial of degree N in terms of the generalized hypergeometric function that depends on p parameters a_1, ..., a_p and q parameters b_1, ..., b_q. The parameters are "generic", possibly complex,…

A series of integral lattices parametrised by integers $k,m,n$ are introduced and investigated, where $n$ is the rank of the lattice, including the root lattices described in a uniform way and unimodular lattices such as the Niemeier…