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Brendan Hassett

Publications and source records attributed to Brendan Hassett.

At least 37 records · Page 2Linked to original sources
arXiv2018

Cremona transformations and derived equivalences of K3 surfaces

We exhibit a Cremona transformation of ${\bf P}^4$ such that the base loci of the map and its inverse are birational to K3 surfaces. The two K3 surfaces are derived equivalent but not isomorphic to each other. As an application, we show that the difference of the two K3 surfaces annihilates the class of the affine line in the Grothendieck ring of varieties.

math.AG↗
arXiv2017

Intersections of three quadrics in $\mathbb{P}^7$

We study rationality properties of smooth complete intersections of three quadrics in $\mathbb{P}^7$. We exhibit a smooth family of such intersections with both rational and non-rational fibers.

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arXiv2016

Cubic fourfolds, K3 surfaces, and rationality questions

This is a survey of the geometry of complex cubic fourfolds with a view toward rationality questions. Topics include classical constructions of rational examples, Hodge structures and special cubic fourfolds, associated K3 surfaces and their geometric interpretations, and connections with rationality and unirationality constructions.

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arXiv2013

Balanced line bundles and equivariant compactifications of homogeneous spaces

Manin's conjecture predicts an asymptotic formula for the number of rational points of bounded height on a smooth projective variety in terms of its global geometric invariants. The strongest form of the conjecture implies certain inequalities among geometric invariants of the variety and of its subvarieties. We provide a general geometric framework explaining these phenomena, via the notion of balanced line bundles, and prove the required inequalities for a large class of equivariant compactifications of homogeneous spaces.

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arXiv2013

Embedding pointed curves in K3 surfaces

We analyze morphisms from pointed curves to K3 surfaces with a distinguished rational curve, such that the marked points are taken to the rational curve, perhaps with specified cross ratios. This builds on work of Mukai and others characterizing embeddings of curves into K3 surfaces via non-abelian Brill-Noether theory. Our study leads naturally to enumerative problems, which we solve in several specific cases. These have applications to the existence of sections of del Pezzo fibrations with prescribed invariants.

math.AG↗
arXiv2013

Quartic del Pezzo surfaces over function fields of curves

We classify quartic del Pezzo surface fibrations over the projective line via numerical invariants, giving explicit examples for small values of the invariants. For generic such fibrations, we describe explicitly the geometry of spaces of sections to the fibration, and mappings to the intermediate Jacobian of the total space. We exhibit examples where these are birational, which has applications to arithmetic questions, especially over finite fields.

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arXiv2011

Spaces of sections of quadric surface fibrations over curves

We consider quadric surface fibrations over curves, defined over algebraically closed and finite fields. Our goal is to understand, in geometric terms, spaces of sections for such fibrations. We analyze varieties of maximal isotropic subspaces in the fibers as P^1-bundles over the discriminant double cover. When the P^1-bundle is suitably stable, we deduce effective estimates for the heights of sections over finite fields satisfying various approximation conditions. We also discuss the behavior of the spaces of sections as the base of the fibration acquires singularities.

math.AG↗