Smoothing algebraic space curves.

*(English)*Zbl 0574.14028
Algebraic geometry, Proc. Conf., Sitges (Barcelona)/Spain 1983, Lect. Notes Math. 1124, 98-131 (1985).

[For the entire collection see Zbl 0551.00002.]

In this paper we review the general question of smoothing singular curves, especially in \({\mathbb{P}}^ 3\). We would like to give necessary and sufficient conditions for a singular (reducible) curve to be smoothable. While this goal seems to be unreachable for the moment, we are able to give two smoothing theorems (4.1) and (4.5) which include old results of Severi, rediscovered recently by Tannenbaum and Ballico and Ellia, as well as Sernesi’s technique of adding 4-secant conics to a curve.

In section 1, we give the statement in deformation theory which will yield our smoothing results (this is the starting point in Sernesi’s work, too). In section 2, we review elementary transformations of the normal bundle of irreducible components of the curve to be smoothed. Section 3 describes the elementary transformations connecting the normal sheaf of a nodal curve to the normal bundle of its smooth irreducible components. Besides our main results, section 4 includes a number of examples. In section 5 we give some examples of nonsmoothable curves, to set the results of the fourth section in perspective. In section 6 we reconsider the question of smoothing ”stick figures” which was discussed at some length by Severi in the famous Anhang G.

In this paper we review the general question of smoothing singular curves, especially in \({\mathbb{P}}^ 3\). We would like to give necessary and sufficient conditions for a singular (reducible) curve to be smoothable. While this goal seems to be unreachable for the moment, we are able to give two smoothing theorems (4.1) and (4.5) which include old results of Severi, rediscovered recently by Tannenbaum and Ballico and Ellia, as well as Sernesi’s technique of adding 4-secant conics to a curve.

In section 1, we give the statement in deformation theory which will yield our smoothing results (this is the starting point in Sernesi’s work, too). In section 2, we review elementary transformations of the normal bundle of irreducible components of the curve to be smoothed. Section 3 describes the elementary transformations connecting the normal sheaf of a nodal curve to the normal bundle of its smooth irreducible components. Besides our main results, section 4 includes a number of examples. In section 5 we give some examples of nonsmoothable curves, to set the results of the fourth section in perspective. In section 6 we reconsider the question of smoothing ”stick figures” which was discussed at some length by Severi in the famous Anhang G.

##### MSC:

14H45 | Special algebraic curves and curves of low genus |

14H10 | Families, moduli of curves (algebraic) |