an:02021087
Zbl 1044.14019
Bonavero, Laurent; Casagrande, Cinzia; Debarre, Olivier; Druel, St??phane
On a conjecture of Mukai
FR
Comment. Math. Helv. 78, No. 3, 601-626 (2003).
00096317
2003
j
14J45 14E30 14M25
Fano varieties; Mori theory; toric varieties
The authors use the following notations: \(X\) -- a Fano variety, \(\rho_X\) -- its Picard number, \(r_X\) -- the index of \(X\), the largest integer \(m\) such that \(K_X = m \cdot L\) holds in the Picard group of \(X\), and \(\iota_X\) -- the pseudo index, the smallest intersection number of the form \((-K_X)\cdot C \) with \(C\) a rational curve in \(X\).
Mukai conjectured that the inequality \(\rho_X(r_X-1) \leq \text{ dim}(X)\) holds. The authors generalize this conjecture to
\[
\rho_X(\iota_X-1) \leq \text{ dim}(X)
\]
and prove it for the following cases:
\(X\) is a Fano variety of dimension \(\leq 4\), \(X\) is a toric variety of dimension \(\leq 7\), and \(X\) is toric and satisfies \(\iota_X \geq \frac{\text{ dim}(X)+3}{3}\).
The proof uses techniques of Mori and from the theory of toric varieties. It uses extremal contractions, chains of rational curves, families of rational curves and the pairing between \(N_{1(X)}\) and \(\text{ Pic}(X)\).
Georg Hein (Berlin)