Smoothing algebraic space curves

Generalized Wahl maps and adjoint line bundles on a general curve

“…In fact, since C C Y is a local complete intersection the first condition implies that the Hubert scheme of Y is smooth at C", while the second says that there are embedded first order deformations of C which smooth the nodes (cf. [HH83]). …”

Points rationnels des courbes génériques de ${\bbfP}\sp 3$. I

“…D'autre part, il est facile de voir que la famille universelle des courbes de bidegré (û, b) contenues dans une quadrique lisse fixée de P 3 admet une section rationnelle si et seulement si û ou b vaut 1 (voir [8] [5], § 1).…”

“…Cependant leurs résultats (en particulier, dans [5], la proposition 1.1, les théorèmes 4.1, 4.5 et leurs preuves) restent valables. Pour tout changement de base Z ->• H^y on note U^ l'image réciproque sur Z de l/Jy.…”

Smoothing curves in 𝑃³ with 𝑝ₐ=1

“…In §2, we prove that a connected, reduced curve in P3 of arithmetic genus 1 is smoothable. The proof is based on a result of Hartshorne and Hirschowitz [1] and a formula for the arithmetic genus of a curve given in [2] (see §1 for its statement).…”

“…In [3] Tannenbaum proved that every connected, reduced curve in P3 of arithmetic genus 0 may be smoothed. Here we prove, using results of Hartshorne and Hirschowitz [1], that every connected, reduced curve in P3 of arithmetic genus 1 is also smoothable.Introduction. Let X be a connected, reduced curve in P3.…”