A₁ del Pezzo surface of degree 6

Abstract

Let S⊂ℙ⁶ be the surface cut out by the intersection of 9 quadrics

$$ \begin{gathered} x_1^2 - x_2 x_4 = x_1 x_5 - x_3 x_4 = x_1 x_3 - x_2 x_5 = x_1 x_6 - x_3 x_5 \hfill \\ = x_2 x_6 - x_3^2 = x_4 x_6 - x_5^2 = x_1^2 - x_2 x_4 + x_5 x_7 \hfill \\ = x_1^2 - x_1 x_2 - x_3 x_7 = x_1 x_3 - x_1 x_5 + x_6 x_7 = 0. \hfill \\ \end{gathered} $$

(5.1)

This is the A₁ del Pezzo surface of degree 6 that was introduced in (2.21). Any line in ℙ⁶ is defined by the intersection of 5 hyperplanes. It is not hard to see that the equations

$$ \left\{ \begin{gathered} x_1 = x_2 = x_3 = x_5 = x_6 = 0, x_1 = x_3 = x_4 = x_5 = x_6 = 0, \hfill \\ x_3 = x_5 = x_6 = x_1 - x_4 = x_1 - x_2 = 0, \hfill \\ \end{gathered} \right. $$

(5.2)

all define lines contained in S. Table 2.6 ensures that these are the only lines contained in S. We set U to be the open subset of S obtained by deleting the lines.