Exceptional collections, and the Néron-Severi lattice for surfaces
Let \(X\) be a smooth projective surface over a field \(k\), which can be chosen of any characteristic. In the present paper, the author tries to investigate when \(X\) has full or numerical exceptional collections of objects of maximal length. Recall that an object \(E\) of \(\mathrm{D}^{\mathrm{b}}(X)\) is exceptional if NEWLINE\[NEWLINE \mathrm{Hom}_{\mathrm{D}^{\mathrm{b}}(X)}(E, E[\ell]) = \begin{cases} k \text{ if }\ell=0\\ 0\text{ if }\ell \neq 0\end{cases}NEWLINE\]NEWLINE A family \(\{E_1, \dots, E_n \}\) of exceptional objects is an exceptional collection if and only if NEWLINE\[NEWLINE \mathrm{Hom}_{\mathrm{D}^{\mathrm{b}}(X)}(E_i, E_j[\ell]) =0\text{ if }i>j\text{ and }\ell \in \mathbb{Z}.NEWLINE\]NEWLINE This collection is full if it generates the category \(\mathrm{D}^{\mathrm{b}}(X)\). The most celebrated example of a full exceptional collection is due to Beilinson: \(\{O, O(1), \ldots, O(n)\}\) is a full exceptional collection on \(\mathbb{P}^n\). These definitions can be weakened as follows: an object \(E\) of \(\mathrm{D}^{\mathrm{b}}(X)\) is numerically exceptional if \(\chi(E, E)=1\). A family \(\{E_1, \dots, E_n \}\) of numerically exceptional objects is a numerically exceptional collection if and only if \(\chi(E_i, E_j) =0\) if \(i>j\). The collection is full if it generates the numerical Grothendieck group of \(X\). If \(X\) is a surface such that \(\chi(\mathcal{O}_X)=1\), the main result of the paper is a necessary and sufficient condition on the Neron-Severi lattice of \(X\) that ensures that \(X\) admits a full numerically exceptional collection. Some consequences of this results are presented.
scientific article; zbMATH DE number 6652656
scientific article