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Some results on embedding of a line in 3-space. (English) Zbl 0782.14011

Let \(K\) be an algebraically closed field and let \(\tau:\mathbb{A}^ 1_ K\hookrightarrow\mathbb{A}^ r_ K\) be the standard embedding given by \(\tau(t)=(t,0,\ldots,0)\). If \(\theta:\mathbb{A}^ 1_ K\hookrightarrow\mathbb{A}^ r_ K\) is another embedding, then we say that \(\theta\) is equivalent to \(\tau\) if there exists a \(K\)-automorphism \(\sigma\) of \(\mathbb{A}^ r_ K\) such that \(\sigma\theta=\tau\). The famous “epimorphism theorem” of S. S. Abhyankar and T. Moh [J. Reine Angew. Math. 276, 148-166 (1975; Zbl 0332.14004)] says that when \(\text{char} K=0\), any embedding of \(\mathbb{A}^ 1_ K\) in \(\mathbb{A}^ 2_ K\) is equivalent to \(\tau\). Moreover, this theorem is not valid when \(\text{char} K>0\) [M. Nagata, “On automorphism group of \(k(x,y)\)”, Lect. Math. 5 (1972; Zbl 0306.14001); p. 39]. – Later, S. S. Abhyankar [in Proc. Int. Symp. Algebraic Geometry, Kyoto 1977, 249-414 (1977; Zbl 0408.14010); conjecture 1, p. 413] conjectured that for \(r\geq 3\), there exist embeddings of \(\mathbb{A}^ 1_ K\) in \(\mathbb{A}^ r_ K\) which are not equivalent to the standard embedding \(\tau\) (when \(\text{char} K=0)\). Z. Jelonek [Math. Ann. 277, 113-120 (1987; Zbl 0611.14010); theorem 1.1] showed that any embedding of \(\mathbb{A}^ 1_ K\) in \(\mathbb{A}^ r_ K\) is equivalent to \(\tau\) for \(r\geq 4\) when \(K=\mathbb{C}\). In fact, his proof is valid for an algebraically closed field \(K\) of any characteristic. Therefore, one is only interested in embeddings of \(\mathbb{A}^ 1_ K\) in \(\mathbb{A}^ 3_ K\).
Conjecture A. Let \(\theta(t)=\bigl(f(t),g(t),h(t)\bigr):A^ 1_ K\hookrightarrow\mathbb{A}^ 3_ K\) be an embedding such that none of the integers \(\deg f\), \(\deg g\), and \(\deg h\) belongs to the semigroup generated by the other two. Then \(\theta\) is not equivalent to the standard embedding \(\tau:\tau(t)=(t,0,0)\).
More specifically:
Conjecture B. The embeddings \(\theta_ n(t)=(t+t^{n+2},t^{n+1},t^ n)\) of \(\mathbb{A}^ 1_ K\) in \(\mathbb{A}^ 3_ K\) are not equivalent to \(\tau\) for \(n\geq 3\).
In this paper we prove (theorem 2.2) that if \(\text{char} K>0\), then \(\theta_ n\) is equivalent to \(\tau\) for all \(n\). This appears to be a little unusual considering the fact that the epimorphism theorem is false in positive characteristic. Moreover, when \(\text{char} K=0\), we show that the embedding \((t+t^ l,t^ m,t^ n)\) is equivalent to \(\tau\), where \(m\equiv 1\pmod n\), \(l\equiv -1\pmod n\) and \(l>n\). Note that \(\theta_ 3\) is a special case of an embedding of this type. – Further, we also show that \(\theta_ 4\) is equivalent to \(\tau\).

MSC:

14E25 Embeddings in algebraic geometry
14H50 Plane and space curves
14J30 \(3\)-folds
14H37 Automorphisms of curves
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References:

[1] Abhyankar, S. S., On the semigroup of a meromorphic curve, (Intl. Symp. on Alg. Geom (Part I). Intl. Symp. on Alg. Geom (Part I), Kyoto (1977)), 249-414
[2] Abhyankar, S. S.; Moh, T. T., Embeddings of the line in the plane, J. Reine Angew. Math., 276, 148-166 (1975) · Zbl 0332.14004
[3] Abhyankar, S. S.; Singh, B., Embeddings of certain curves in the affine plane, Amer. J. Math., 100, 99-175 (1978) · Zbl 0383.14007
[4] Craighero, P. C., Osservazioni sopra alcuni esempi di curve dello spazio \(A_k^3\) isomorphe a rette, Boll. Un. Mat. Ital. B(6), 1, 1199-1216 (1982) · Zbl 0504.14024
[5] Craighero, P. C., About Abhyankar’s Conjectures on space lines, (Rend. Sem. Mat. Univ. Padova, 74 (1985)), 115-121 · Zbl 0594.14025
[6] Craighero, P. C., A result on \(m\)-flats in \(A_k^{n\) · Zbl 0601.14010
[7] Craighero, P. C., A remark on Abhyankar’s space lines, (Rend. Sem. Mat. Univ. Padova, 80 (1988)), 87-93 · Zbl 0702.14027
[8] Jelonek, Z., The extension of regular and rational embeddings, Math. Ann., 277, 113-120 (1987) · Zbl 0611.14010
[9] Nagata, M., On automorphism group of \(k[X, Y]\), (Lec. Math., Vol. 5 (1972), Kyoto Univ: Kyoto Univ Kinokuniya, Tokyo) · Zbl 0306.14001
[11] Russell, P.; Sathaye, A., On finding and cancelling variables in \(k[X, Y, Z]\), J. Algebra, 57, 151-166 (1979) · Zbl 0411.13011
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