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Alexander Perry

Publications and source records attributed to Alexander Perry.

28 records · Page 2Linked to original sources
arXiv2019

Categorical cones and quadratic homological projective duality

We introduce the notion of a categorical cone, which provides a categorification of the classical cone over a projective variety, and use our work on categorical joins to describe its behavior under homological projective duality. In particular, our construction provides well-behaved categorical resolutions of singular quadrics, which we use to obtain an explicit quadratic version of the main theorem of homological projective duality. As applications, we prove the duality conjecture for Gushel-Mukai varieties, and produce interesting examples of conifold transitions between noncommutative and honest Calabi-Yau threefolds.

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arXiv2018

Hochschild cohomology and group actions

Given a finite group action on a (suitably enhanced) triangulated category linear over a field, we establish a formula for the Hochschild cohomology of the category of invariants, assuming the order of the group is coprime to the characteristic of the base field. The formula shows that the cohomology splits canonically with one summand given by the invariant subspace of the Hochschild cohomology of the original category. We also prove that Serre functors act trivially on Hochschild cohomology, and combine this with our formula to give a useful mechanism for computing the Hochschild cohomology of fractional Calabi-Yau categories.

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arXiv2018

Noncommutative homological projective duality

We generalize Kuznetsov's theory of homological projective duality to the setting of noncommutative algebraic geometry. Simultaneously, we develop the theory over general base schemes, and remove the usual smoothness, properness, and transversality hypotheses.

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arXiv2018

Intersections of two Grassmannians in $\mathbf{P}^9$

We study the intersection of two copies of $\mathrm{Gr}(2,5)$ embedded in $\mathbf{P}^9$, and the intersection of the two projectively dual Grassmannians in the dual projective space. These intersections are deformation equivalent, derived equivalent Calabi-Yau threefolds. We prove that generically they are not birational. As a consequence, we obtain a counterexample to the birational Torelli problem for Calabi-Yau threefolds. We also show that these threefolds give a new pair of varieties whose classes in the Grothendieck ring of varieties are not equal, but whose difference is annihilated by a power of the class of the affine line. Our proof of non-birationality involves a detailed study of the moduli stack of Calabi-Yau threefolds of the above type, which may be of independent interest.

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arXiv2017

Derived categories of Gushel-Mukai varieties

We study the derived categories of coherent sheaves on Gushel-Mukai varieties. In the derived category of such a variety, we isolate a special semiorthogonal component, which is a K3 or Enriques category according to whether the dimension of the variety is even or odd. We analyze the basic properties of this category using Hochschild homology, Hochschild cohomology, and the Grothendieck group. We study the K3 category of a Gushel-Mukai fourfold in more detail. Namely, we show that this category is equivalent to the derived category of a K3 surface for a certain codimension 1 family of rational fourfolds, and to the K3 category of a birational cubic fourfold for a certain codimension 3 family. The first of these results verifies a special case of a duality conjecture which we formulate. We discuss our results in the context of the rationality problem for Gushel-Mukai varieties, which was one of the main motivations for this work.

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arXiv2016

Rationality does not specialize among terminal fourfolds

We show that rationality does not specialize in flat projective families of complex fourfolds with terminal singularities. This answers a question of Totaro, who established the analogous result in all dimensions greater than 4.

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arXiv2016

Rational points on twisted K3 surfaces and derived equivalences

Using a construction of Hassett--Várilly-Alvarado, we produce derived equivalent twisted K3 surfaces over $\mathbb{Q}$, $\mathbb{Q}_2$, and $\mathbb{R}$, where one has a rational point and the other does not. This answers negatively a question recently raised by Hassett and Tschinkel.

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arXiv2015

Derived categories of cyclic covers and their branch divisors

Given a variety $Y$ with a rectangular Lefschetz decomposition of its derived category, we consider a degree $n$ cyclic cover $X \to Y$ ramified over a divisor $Z \subset Y$. We construct semiorthogonal decompositions of $\mathrm{D^b}(X)$ and $\mathrm{D^b}(Z)$ with distinguished components $\mathcal{A}_X$ and $\mathcal{A}_Z$, and prove the equivariant category of $\mathcal{A}_X$ (with respect to an action of the $n$-th roots of unity) admits a semiorthogonal decomposition into $n-1$ copies of $\mathcal{A}_Z$. As examples we consider quartic double solids, Gushel-Mukai varieties, and cyclic cubic hypersurfaces.

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arXiv2009

Resolving toric varieties with Nash blow-ups

It is a long-standing question whether an arbitrary variety is desingularized by finitely many normalized Nash blow-ups. We consider this question in the case of a toric variety. We interpret the normalized Nash blow-up in polyhedral terms, show how continued fractions can be used to give an affirmative answer for a toric surface, and report on a computer investigation in which over a thousand 3- and 4-dimensional toric varieties were successfully resolved.

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