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Helices on some Fano threefolds: Constructivity of semiorthogonal bases of \(K_ 0\). (English) Zbl 0845.14012

The exceptional bundles were introduced on \(\mathbb{P}_2\) by the reviewer and J. Le Potier [Ann. Sci. Éc. Norm. Supér., IV. Ser. 18, 193-243 (1985; Zbl 0586.14007)], and generalized to \(\mathbb{P}_n\) by A. L. Gorodentsev and A. N. Rudakov [Duke Math. J. 54, 115-130 (1987; Zbl 0646.14014)], where the important notion of helix appears. Exceptional bundles were also studied on other varieties [cf. A. N. Rudakov, in: Helices and vector bundles, Proc. Semin. Rudakov, Lond. Math. Soc. Lect. Note Ser. 148, 1-6 (1990; Zbl 0721.14011)]. On some varieties \(X\) such as projective spaces, one considers particular sequences \((E_i)_{i \in \mathbb{Z}}\) of exceptional bundles, called helices, which are periodic, i.e. if \(d = \text{rank} (K_0 (X))\), then we have, for every integer \(i\), \(E_{i - d} \simeq E_i \otimes K_X\). A sequence \((E_i, E_{i + 1}, \dots, E_{i + d - 1})\) is called a foundation of the helix \((E_i)_{i \in \mathbb{Z}}\).
These foundations, viewed as sequences of elements of the Grothendieck group \(K_0 (X)\), are semi-orthogonal bases. This means the following: Let \(\chi \) be the canonical bilinear form on \(K_0 (X)\) (defined by \(\chi ([E], [F]) = \chi (E \otimes F)\) for all vector bundles \(E\), \(F\) on \(X)\). Let \(e_j = [E_{i + j}] \in K_0 (X)\) for \(0 \leq j < d\). Then we have \(\chi (e_{j'}, e_j) = 0\) if \(j' > j\), \(\chi (e_j, e_j) = 1\).
It is possible, using mutations, to obtain new helices from a given one. The translation of this is the mutation of semi-orthogonal bases of \(K_0 (X)\): If \((e_i)_{0 \leq i < d}\) is a semi-orthogonal basis of \(K_0 (X)\) and \(0 \leq j < d\), then the mutation of \((e_i)\) at \(j\) is the semi-orthogonal basis \((e_i')\) defined by \(e_i'= e_i \) if \(i \neq j,\) \(j + 1 \pmod d\), \(e_j'= \chi (e_j, e_{j + 1}) e_j - e_{j + 1}\), \(e_{j + 1}'= e_j\). In this paper the author proves the following: Let \(X\) be \(\mathbb{P}_3\), a smooth 3-dimensional quadric or one of the Fano 3-folds \(V_5\) or \(V_{22}\). Then, up to changing signs of some elements, every semi-orthogonal basis of \(K_0 (X)\) can be obtained by successive mutations from some elementary semi-orthogonal bases, called canonical bases. These canonical bases are, in the case of \(\mathbb{P}_3\), given by \(\bigl( {\mathcal O} \otimes {\mathcal I}^n_p, {\mathcal O} (1) \otimes {\mathcal I}^n_p, {\mathcal O} (2) \otimes {\mathcal I}^n_p, {\mathcal O} (3) \otimes {\mathcal I}^n_p \bigr),\) where \({\mathcal I}_p\) is the ideal sheaf of the point \(p \in \mathbb{P}_3\). There are similar results for \(Q\) and \(V_5\). For \(V_{22}\) there is only a numerical description.
To prove these results the author finds and uses some equations verified by the invariants of the exceptional bundles in a foundation, called Markov-type equations. An interesting corollary of the proofs is that in a foundation of a helix on \(\mathbb{P}_3\), \(Q\), \(V_5\) or \(V_{22}\), the exceptional bundles are ordered by slopes.

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J45 Fano varieties
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14J30 \(3\)-folds
19A49 \(K_0\) of other rings
19E08 \(K\)-theory of schemes
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References:

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