Given a scheme Y equipped with a collection of globally generated vector bundles $$E_1, \ldots , E_n$$
 , we study the universal morphism from Y to a fine moduli space $${\mathcal {M}}(E)$$
 of cyclic modules over the endomorphism algebra of $$E:={\mathcal {O}}_Y\oplus E_1\oplus \cdots \oplus E_n$$
 . This generalises the classical morphism to the linear series of a basepoint-free line bundle on a scheme. We describe the image of the morphism and present necessary and sufficient conditions for surjectivity in terms of a recollement of a module category. When the morphism is surjective, this gives a fine moduli space interpretation of the image, and as an application we show that for a small, finite subgroup $$G\subset {{\mathrm{GL}}}(2,\mathbb {k})$$
 , every sub-minimal partial resolution of $${\mathbb {A}}^2_\mathbb {k}/G$$
 is isomorphic to a fine moduli space $${\mathcal {M}}(E_C)$$
 where $$E_C$$
 is a summand of the bundle E defining the reconstruction algebra. We also consider applications to Gorenstein affine threefolds, where Reid’s recipe sheds some light on the classes of algebra from which one can reconstruct a given crepant resolution.

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Multigraded linear series and recollement

We prove the existence of tilting objects on some global quotient stacks. As a consequence, we provide further evidence for a conjecture on the Rouquier dimension of derived categories formulated by Orlov.

Tilting objects on some global quotient stacks

We combine the Bondal-Uehara method for producing exceptional collections on toric varieties with a result of the first author and Favero to expand the set of varieties satisfying Orlov's Conjecture on derived dimension

The toric Frobenius morphism and a conjecture of Orlov

We prove the existence of tilting objects on some global quotient stacks. As a consequence we provide further evidence for a conjecture on the Rouquier dimension of derived categories formulated by Orlov.

We begin a systematic investigation of derived categories of smooth projective toric varieties defined over an arbitrary base field. We show that, in many cases, toric varieties admit full exceptional collections. Examples include all toric surfaces, all toric Fano 3-folds, some toric Fano 4-folds, the generalized del Pezzo varieties of Voskresenskii and Klyachko, and toric varieties associated to Weyl fans of type $A$. Our main technical tool is a completely general Galois descent result for exceptional collections of objects on (possibly non-toric) varieties over non-closed fields.

On derived categories of arithmetic toric varieties

Tilting bundles on toric Fano fourfolds

This paper constructs tilting bundles obtained from full strong exceptional collections of line bundles on all smooth $4$-dimensional toric Fano varieties. The tilting bundles lead to a large class of explicit Calabi-Yau-$5$ algebras, obtained as the corresponding rolled-up helix algebra. A database of the full strong exceptional collections can be found in the package QuiversToricVarieties for the computer algebra system Macaulay2.

Nathan Prabhu-Naik

Papers

Tilting bundles on toric Fano fourfolds

Tilting Bundles on Toric Fano Fourfolds