Fourier-Mukai partners of elliptic ruled surfaces. (English) Zbl 1391.14031
Consider a smooth projective variety \(X\) over \(\mathbb{C}\), and denote by \(D(X)\) the bounded derived category of coherent sheaves on \(X\). If the canonical bundle of \(X\) is either ample or anti-ample, then \(D(X)\) determines \(X\) up to isomorphisms, but this is not true in general. So it makes sense to study non-trivial instances of the following
Definition. Two varieties \(X\) and \(Y\) as above are called Fourier-Mukai partners if \(D(X)\) and \(D(Y)\) are equivalent.
In this paper, the author study the case of \(\mathbb{P}^1\)-bundle \(S:=\mathbb{P}(\mathcal{E})\) over an elliptic curve \(E\), where \(\mathcal{E}\) is a normalized locally free sheaf of rank \(2\) over \(E\). Denote by \(\text{FM}(X)\) the classes of Fourier-Mukai partners of \(X\) up to isomorphisms. The main result is the following:
Theorem. When \(S\) has at least one non-trivial Fourier-Mukai partner, that is \(|\text{FM}(S)|\neq 1\), then the sheaf \(\mathcal{E}\) is equal to \(\mathscr{O}_E \oplus \mathcal{L}\) for a certain line bundle \(\mathcal{L} \in \text{Pic}^0(E)\) of order greater than \(4\). Moreover we have the following description \[ \text{FM}(S)=\{\mathbb{P}(\mathscr{O}_E \oplus \mathcal{L}^i)|i \in (\mathbb{Z}/m\mathbb{Z})^*\}/\cong. \]
As an application, the group of autoequivalences of a \(\mathbb{P}^1\) bundle \(\mathbb{P}(\mathscr{O}_E \oplus \mathcal{L})\) over an elliptic curve \(E\), where \(\mathcal{L}\) is contained in the sets of points of \(E\) of order \(m > 4\). This is achieved by applying a result of [H. Uehara, Algebr. Geom. 3, No. 5, 543–577 (2016; Zbl 1376.14024)].
Definition. Two varieties \(X\) and \(Y\) as above are called Fourier-Mukai partners if \(D(X)\) and \(D(Y)\) are equivalent.
In this paper, the author study the case of \(\mathbb{P}^1\)-bundle \(S:=\mathbb{P}(\mathcal{E})\) over an elliptic curve \(E\), where \(\mathcal{E}\) is a normalized locally free sheaf of rank \(2\) over \(E\). Denote by \(\text{FM}(X)\) the classes of Fourier-Mukai partners of \(X\) up to isomorphisms. The main result is the following:
Theorem. When \(S\) has at least one non-trivial Fourier-Mukai partner, that is \(|\text{FM}(S)|\neq 1\), then the sheaf \(\mathcal{E}\) is equal to \(\mathscr{O}_E \oplus \mathcal{L}\) for a certain line bundle \(\mathcal{L} \in \text{Pic}^0(E)\) of order greater than \(4\). Moreover we have the following description \[ \text{FM}(S)=\{\mathbb{P}(\mathscr{O}_E \oplus \mathcal{L}^i)|i \in (\mathbb{Z}/m\mathbb{Z})^*\}/\cong. \]
As an application, the group of autoequivalences of a \(\mathbb{P}^1\) bundle \(\mathbb{P}(\mathscr{O}_E \oplus \mathcal{L})\) over an elliptic curve \(E\), where \(\mathcal{L}\) is contained in the sets of points of \(E\) of order \(m > 4\). This is achieved by applying a result of [H. Uehara, Algebr. Geom. 3, No. 5, 543–577 (2016; Zbl 1376.14024)].
Reviewer: Riccardo Moschetti (Stavanger)
MSC:
| 14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |
| 14J27 | Elliptic surfaces, elliptic or Calabi-Yau fibrations |
| 18E30 | Derived categories, triangulated categories (MSC2010) |
Keywords:
Fourier-Mukai partners; elliptic surfaces; elliptic fibrations; elliptic ruled surfaces; autoequivalences; group of autoequivalencesCitations:
Zbl 1376.14024References:
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