Abstract
We survey the period maps of some Fano varieties and the geometry of their spaces of curves of low genera and degrees.
Mathematics Subject Classification codes (2010): 14C05, 14C30, 14C34, 14D20, 14E05, 14E08, 14E20, 14H10, 14J10, 14J30, 14J35, 14J45, 14J60, 14J70, 14M20, 14M22, 14N25
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Notes
- 1.
Let Δ : Z → Z ×Z be the diagonal embedding and let Δ(Z)(2) ⊂ Z ×Z be the closed subscheme defined by the sheaf of ideals \(\mathcal{I}_{\Delta (Z)}^{2}\). Since \(\mathcal{I}_{\Delta (Z)}/\mathcal{I}_{\Delta (Z)}^{2} \simeq \Omega _{Z}\); we have an exact sequence
$$\displaystyle{0 \rightarrow \Delta _{{\ast}}\Omega _{Z} \rightarrow \mathcal{O}_{\Delta {(Z)}^{(2)}} \rightarrow \Delta _{{\ast}}\mathcal{O}_{Z} \rightarrow 0.}$$If \(\mathcal{F}\) is a locally free sheaf on Z, we obtain an exact sequence
$$\displaystyle{0 \rightarrow \mathcal{F}\otimes \Omega _{Z} \rightarrow p_{1{\ast}}(p_{2}^{{\ast}}(\mathcal{F}\otimes \mathcal{O}_{ \Delta {(Z)}^{(2)}})) \rightarrow \mathcal{F}\rightarrow 0,}$$hence an extension class \(At_{\mathcal{F}}\in Ex{t}^{1}(\mathcal{F},\mathcal{F}\otimes \Omega _{Z})\). The same construction can be extended to any coherent sheaf on Z by working in the derived category (Illusie).
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Acknowledgements
O. Debarre is part of the project VSHMOD-2009 ANR-09-BLAN-0104-01. These are notes from a talk given at the conference “Geometry Over Non-Closed Fields” funded by the Simons Foundation, whose support is gratefully acknowledged.
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Debarre, O. (2013). Curves of Low Degrees on Fano Varieties. In: Bogomolov, F., Hassett, B., Tschinkel, Y. (eds) Birational Geometry, Rational Curves, and Arithmetic. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6482-2_6
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