Space curves, X-ranks and cuspidal projections

Let $$X\subset \mathbb {P}^3$$ X ⊂ P 3 be an integral and non-degenerate curve. We say that $$q\in \mathbb {P}^3\setminus X$$ q ∈ P 3 \ X has X-rank 3 if there is no line $$L\subset \mathbb {P}^3$$ L ⊂ P 3 such that $$q\in L$$ q ∈ L and $$\#(L\cap X)\ge 2$$ # ( L ∩ X ) ≥ 2 . We prove that for all hyperelliptic curves of genus $$g\ge 5$$ g ≥ 5 there is a degree $$g+3$$ g + 3 embedding $$X\subset \mathbb {P}^3$$ X ⊂ P 3 with exactly $$2g+2$$ 2 g + 2 points with X-rank 3 and another embedding without points with X-rank 3 but with exactly $$2g+2$$ 2 g + 2 points $$q\in \mathbb {P}^3$$ q ∈ P 3 such that there is a unique pair of points of X spanning a line containing q. We also prove the non-existence of points of X-rank 3 for general curves of bidegree (a, b) in a smooth quadric (except in known exceptional cases) and we give lower bounds for the number of pairs of points of X spanning a line containing a fixed $$q\in \mathbb {P}^3\setminus X$$ q ∈ P 3 \ X . For all integers $$g\ge 5$$ g ≥ 5 , $$x\ge 0$$ x ≥ 0 we prove the existence of a nodal hyperelliptic curve X with geometric genus g, exactly x nodes, $$\deg (X) = x+g+3$$ deg ( X ) = x + g + 3 and having at least $$x+2g+2$$ x + 2 g + 2 points of X-rank 3.

Degenerate Curve Bifurcations in 3D Linear Symmetric Tensor Fields

Abstract3D symmetric tensor fields have a wide range of applications in medicine, science, and engineering. The topology of tensor fields can provide key insight into their structures. In this paper we study the number of possible topological bifurcations in 3D linear tensor fields. Using the linearity/planarity classification and wedge/trisector classification, we explore four types of bifurcations that can change the number and connectivity in the degenerate curves as well as the number and location of transition points on these degenerate curves. This leads to four types of bifurcations among nine scenarios of 3D linear tensor fields.

On the Hilbert scheme of smooth curves in ℙ4 of degree d = g + 1 and genus g with negative Brill–Noether number

We denote by [Formula: see text] the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree [Formula: see text] and genus [Formula: see text] in [Formula: see text]. In this paper, we show that for low genus [Formula: see text] outside the Brill–Noether range, the Hilbert scheme [Formula: see text] is non-empty whenever [Formula: see text] and irreducible whose only component generically consists of linearly normal curves unless [Formula: see text] or [Formula: see text]. This complements the validity of the original assertion of Severi regarding the irreducibility of [Formula: see text] outside the Brill–Nother range for [Formula: see text] and [Formula: see text].

A Vanishing Theorem for the Twisted Normal Bundle of Curves in ,

We prove the existence of a smooth and non-degenerate curve $X\subset \mathbb{P}^{n}$, $n\geqslant 8$, with $\deg (X)=d$, $p_{a}(X)=g$, $h^{1}(N_{X}(-1))=0$, and general moduli for all $(d,g,n)$ such that $d\geqslant (n-3)\lceil g/2\rceil +n+3$. It was proved by C. Walter that, for $n\geqslant 4$, the inequality $2d\geqslant (n-3)g+4$ is a necessary condition for the existence of a curve with $h^{1}(N_{X}(-1))=0$.