On the Halphen transform of algebraic space curves

ABSTRACT

The Halphen transform of a plane curve is the curve obtained by intersecting the tangent lines of the curve with the corresponding polar lines with respect to some conic. This transform was introduced by Halphen as a branch desingularization method in [5] and has also been studied in [2, 8]. We extend this notion to the Halphen transform of a space curve and study several of its properties (birationality, degree, rank, class, desingularization).

KEYWORDS:

• Degree
• desingularization
• Halphen’s tranform
• space curve

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• 14J99
• 14H50
• 14E05
• 14N05
• 14N10

Notes

¹Here, nonsingular here that dχ(m)∈W^∨ is nonzero.

²Here, nonsingular means that $Vect\left(d{\chi }^{\left(1\right)}\right(m),\dots ,d{\chi }^{\left(I\right)}(m\left)\right)\subset W^{\vee }$ has dimension 2.

³Recall that, in coordinates, ${\wedge }^2\left(uv\right)=\left(\begin{array}{c}{u}_1{v}_2-u_2{v}_1\\ u_1{v}_3-u_3{v}_1\\ u_1{v}_4-u_4{v}_1\\ u_2{v}_3-u_3{v}_2\\ u_2{v}_4-u_4{v}_2\\ u_3{v}_4-u_4{v}_3\end{array}\right)\in {ℂ}^6$, for any u, v∈W.

⁴With the classical notation $⟨A,B⟩={\sum }_{i=1}^6{a}_i{b}_i$ for any $A=(a_1,\dots ,a_6)$ and $B=(b_1,\dots ,b_6)$ in ℂ⁶.

⁵If there exists an irreducible curve 𝒞₂ contained in 𝒞₁ such that $\forall P\in {\mathcal{𝒞}}_2\setminus Sing{\mathcal{𝒞}}_1$, ${\mathcal{𝒯}}_P\mathcal{𝒞}\cap {\mathcal{𝒯}}_{P_0}\mathcal{𝒞}\ne \varnothing$, then either 𝒞₂ is a line that intersects ${\mathcal{𝒯}}_{P_0}{\mathcal{𝒞}}_1$ or 𝒞₂ is a curve which is not a line and which is contained in a plane that contains $T_{P_0}\mathcal{𝒞}$. Since the set of irreducible plane curves contained in 𝒞₁ is finite and since 𝒞 is not a plane curve, our assumption on P₀∈𝒞 holds generically.

⁶If a nonsingular point P of 𝒞₁ is contained in $V(F_x,F_t)$, then ${\mathcal{𝒯}}_{P_0}{\mathcal{𝒞}}_1\subset {\mathcal{𝒯}}_P\left(V\right(F\left)\right)$ and so ${\mathcal{𝒯}}_{P_0}{\mathcal{𝒞}}_1\cap {\mathcal{𝒯}}_P{\mathcal{𝒞}}_1\ne \varnothing$.

⁷Indeed, let 𝒜 be the set of planes ℋ⊂ℙ³ containing P₀ and such that there exist P[x:y:z:t]∈V(F,G)∩ℋ and $\tilde{P}[\tilde{x}:ỹ:\tilde{z}:\tilde{t}]\in V(F,F_x,F_t)\cap \mathcal{\mathscr{H}}$ with $[z:t]=[\tilde{z}:\tilde{t}]$ (so $t\tilde{z}-z\tilde{t}=0$) and $(P,\tilde{P})\ne (P_0,P_0)$. Since the set of such couples $(P,\tilde{P})$ with $P\ne \tilde{P}$ has dimension 1 and since the set of such couples $(P,\tilde{P})$ with P = P₀ is finite, we obtain that dim𝒜≤1.