Smoothing curves in 𝑃³ with 𝑝ₐ=1

Abstract

ABSTRACT. 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. We say that X is smoothable if there exists a flat family of curves X( in P3, whose general member Xt is smooth and whose special member Xo is X. 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).I wish to thank Dr. S. Xambó for many valuable suggestions.Notations, (i) All schemes will be projective algebraic defined over a fixed alge-