Some remarks on smooth projective varieties of small degree and codimension

Abstract The purpose of this note is twofold. First, we give a quick proof of Ballico–Chiantini’s theorem stating that a Fano or Calabi–Yau variety of dimension at least 4 in codimension 2 is a complete intersection. Second, we improve Barth–Van de Ven’s result asserting that if the degree of a smooth projective variety of dimension n is less than approximately $$0.63 \cdot n^{1/2},$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0.63</mml:mn> <mml:mo>·</mml:mo> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> then it is a complete intersection. We show that the degree bound can be improved to approximately $$0.79 \cdot n^{2/3}.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0.79</mml:mn> <mml:mo>·</mml:mo> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>/</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:mo>.</mml:mo> </mml:mrow> </mml:math>