We consider the Hilbert scheme $H(d,g)$ of space curves $C$ with homogeneous ideal $I\left(C\right):=H_{*}^0\left({ℐ}_C\right)$ and Rao module $M:=H_{*}^1\left({ℐ}_C\right)$. By taking suitable generizations (deformations to a more general curve) $C^{\prime }$ of $C$, we simplify the minimal free resolution of $I\left(C\right)$ by e.g making consecutive free summands (ghost-terms) disappear in a free resolution of $I\left(C^{\prime }\right)$. Using this for Buchsbaum curves of diameter one ($M_v\ne 0$ for only one $v$), we establish a one-to-one correspondence between the set $𝒮$ of irreducible components of $H(d,g)$ that contain $\left(C\right)$ and a set of minimal...

This paper studies space curves $C$ of degree $d$ and arithmetic genus $g$, with homogeneous ideal $I$ and Rao module $M={\mathrm{H}}_{*}^1\left(\tilde{I}\right)$, whose main results deal with curves which satisfy ${}_0{\mathrm{Ext}}_R^2(M,M)=0$ (e.g. of diameter, $\mathrm{diam}M\le 2$). For such curves we find necessary and sufficient conditions for unobstructedness, and we compute the dimension of the Hilbert scheme, $\mathrm{H}(d,g)$, at $\left(C\right)$ under the sufficient conditions. In the diameter one case, the necessary and sufficient conditions coincide, and the unobstructedness of $C$ turns out to be equivalent to the...

The ring of projective invariants of $n$ ordered points on the projective line is one of the most basic and earliest studied examples in Geometric Invariant Theory. It is a remarkable fact and the point of this paper that, unlike its close relative the ring of invariants of $n$ unordered points, this ring can be completely and simply described. In 1894 Kempe found generators for this ring, thereby proving the First Main Theorem for it (in the terminology introduced by Weyl). In this paper we compute...

Given a real analytic vector field tangent to a hypersurface $V$ with an algebraically isolated singularity we introduce a relative Jacobian determinant in the finite dimensional algebra $\frac{\mathbf{B}}{{\mathrm{Ann}}_{\mathbf{B}}\left(h\right)}$ associated with the singularity of the vector field on $V$. We show that the relative Jacobian generates a 1-dimensional non-zero minimal ideal. With its help we introduce a non-degenerate bilinear pairing, and its signature measures the size of this point with sign. The signature satisfies a law of conservation of...

It is known that it is sufficient to consider in the Jacobian Conjecture only polynomial mappings of the form $F(x₁,...,x_n)=x-H\left(x\right):=(x₁-H₁(x₁,...,x_n),...,x_n-H_n(x₁,...,x_n))$, where $H_j$ are homogeneous polynomials of degree 3 with real coefficients (or $H_j=0$), j = 1,...,n and H’(x) is a nilpotent matrix for each $x=(x₁,...,x_n)\in {ℝ}^n$. We give another proof of Yu’s theorem that in the case of non-negative coefficients of H the mapping F is a polynomial automorphism, and we moreover prove that in that case $deg{F}^{-1}\le {\left(degF\right)}^{indF-1}$, where $indF:=maxind{H}^{\text{'}}\left(x\right):x\in {ℝ}^n$. Note that the above inequality is not true when the coefficients of...