For any curve \(C\) in \(\mathbb{P}^3\) and \(k\geq 0\) we have the canonical mappings \[ \sigma_k:H^0 \bigl({\mathcal I}_C(k) \bigr)\otimes H^0\bigl({\mathcal O}_{ \mathbb{P}^3} (1)\bigr)\to H^0\bigl({\mathcal I}_C(k+1) \bigr). \] If all the \(\sigma_k\)’s are of maximal rank, then \(C\) (or \({\mathcal I}_C)\) is said to be minimally generated. A maximal rank minimally generated curve is said to be naturally generated. For a minimally generated curve it is possible to compute the number, for each degree, of elements in a minimal system of generators for its saturated homogeneous ideal.
The main result of this paper (which assumes the characteristic 0 hypothesis) is the following:
Theorem 1: The generic rational curve of degree \(d\) in \(\mathbb{P}^3\) is minimally generated for \(d\neq 5\) and \(d\leq 73\).
The result deals with a conjecture due to André Hirschowitz: For any \(g\geq 0\), there exists \(d(g)\) such that for \(d\geq d(g)\) the generic curve of some irreducible component of the Hilbert scheme \(H_{d,g}\) is naturally generated.
Similar questions have been raised for subvarieties of every dimension of \(\mathbb{P}^n\), and in 0-dimensional case a relevant work has been done [see A. Hirschowitz and C. Simpson, Invent. Math. 126, No. 3, 467-503 (1996; Zbl 0877.14035) and D. Eisenbud and S. Popescu, Invent. Math. 136, No. 2, 419-449 (1999; Zbl 0943.13011)]. The main techniques of this paper is the “Méthode d’Horace” [see A. Hirschowitz, Manuscr. Math. 50, 337-388 (1985; Zbl 0571.14002)], recalled at the beginning of section 4, combined with elementary transformations [see M. Maruyama in: Algebraic geometry, Proc. Int. Conf., La Rábida 1981, Lect. Notes Math. 961, 241-266 (1982; Zbl 0505.14009)]; in particular two of them are described in section 1 and allow to manipulate points in \(\mathbb{P}^3\), especially for low degrees. Section 3 is devoted to prove that theorem 1 follows from the statements \(R(k)\), \(k\leq 18\), which, roughly speaking, say that the restriction maps \[ H^0\bigl( {\mathcal O}_{\mathbb{P}( \Omega_3)}(1) \otimes\pi^*{\mathcal O}_{\mathbb{P}^3} (k+1)\bigr)\to H^0\bigl({\mathcal O}_{\mathbb{P} (\Omega_3)} (1)\otimes\pi^* {\mathcal O}_{\mathbb{P}^3} (k+1)|_Z \bigr) \] are bijective for a particular configuration \(Z\), pull-back of a rational curve and lines together with points of \(\mathbb{P}(\Omega^3)= \mathbb{P}(\Omega_{\mathbb{P}^3})\). The statements \(R(k)\) are proved in section 4, using some technical results on blowing-ups of section 2.
For the entire collection see [Zbl 0913.00044].