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‎Bl2PPn.md‎

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## $\mathrm{Bl}_2\PP^n$: the blow-up of $\PP^n$ at 2 points.
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All varieties in this class have the same secondary fan, but as the multiplicity of the Cox degrees increases, the $\Theta$-collection expands, and with it the Ext table exhibits more complex behavior. A result of Michalek ([1009.0821](http://arxiv.org/abs/1009.0821v3)) implies that when $n>20$, such varieties do not have strongly exceptional collections of line bundles.
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All varieties in this class have the same secondary fan, but as the multiplicity of the Cox degrees increases, the $\Theta$-collection expands, and with it the Ext table exhibits more complex behavior. Michalek proved in [[1009.0821]](http://arxiv.org/abs/1009.0821v3) that when $n>20$, such varieties do not have strongly exceptional collections of line bundles.
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{% include table.toric.html class='Bl2PPn' row_limit=1 varieties=site.data.Bl2PPn %}
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‎HillePerling.md‎

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## Hille--Perling's Counterexample to King's Conjecture
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We begin with Hirzebruch surface of type 2 and consider 3 consecutive blow-ups.
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We begin with Hirzebruch surface of type 2 and consider 3 consecutive blow-ups. Hille and Perling showed in [[arxiv:math/0602258]](https://arxiv.org/abs/math/0602258v2) that this toric variety does not have a strong exceptional collection of line bundles.
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{% include table.toric.html class='HillePerling' row_limit=1 varieties=site.data.HillePerling %}
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