Beitraege zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 34 (1993), No. 2, 151-156.

Representation of the Hirzebruch-Kleinschmidt Varieties by Quadrics

Guenter Ewald and Alexa Schmeinck

Abstract.

Let $X_n (a_1, \ldots, a_k)$ be an $n$-dimensional smooth projective
toric variety which Kleinschmidt [5] introduced as a
higher-dimensional generalization of Hirzebruch surfaces [3]. We
show that for 
$r : = n - k + \sum_{i=1}^k
              {a_i  + 1 +  n-k \choose
                           n-k }$
there is a projective embedding 
$\phi : X_n (a_1,\ldots,a_k) \hookrightarrow
          {\bf P}^r$
such that the ideal of $\phi (X_n (a_1,\ldots,a_k))$ is
generated by quadratic binomials.