David Stapleton
dajost@umich.edu
CV
teaching. Algebraic Geometry II
previous courses. F21Calc1·S20Calc3·F20Calc3·W20Calc4·F19Calc3·Aug19Alg2·F18Calc2·W18Alg1·F17Alg1
bio.
I am a postdoc in algebraic geometry at the University of Michigan. I was previously at UC San Diego and I got my PhD from Stony Brook University.
publications.
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arxiv 
Higher index Fano varieties with finitely many birational automorphisms,
joint with Nathan Chen. submitted (2021). 
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arxiv abst 
Rational endomorphisms of Fano hypersurfaces,
joint with Nathan Chen. submitted (2021). 
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arxiv abst 
A direct proof that toric rank 2 bundles on projective space split.
Mathematica Scandinavica (2020). 
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arxiv abst 
Maximal Chow constant and cohomologically constant fibrations,
joint with Kristin DeVleming. Commun. in Contemporary Math. (2020). 
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arxiv abst 
Fano hypersurfaces with arbitrarily large degrees of irrationality, joint with Nathan Chen. Forum of Math., Sigma (2020). 
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arxiv abst 
The degree of irrationality of hypersurfaces in various Fano varieties, joint with Brooke Ullery. Manuscripta Mathematica (2019). 
The degree of irrationality of very general hypersurfaces in some homogeneous spaces.
PhD thesis, Stony Brook Univ. (2017). 

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arxiv abst 
The tangent space of the punctual Hilbert scheme, joint with Dori Bejleri. Mich. Math Journal (2017). 
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arxiv abst 
Geometry and stability of tautological bundles on Hilbert schemes of points.
Algebra and Number Theory (2016). 
abstract.
We show that the degrees of rational endomorphisms of very general complex Fano and CalabiYau hypersurfaces satisfy certain congruence conditions by specializing to characteristic p.
abstract.
The point of this paper is to give a short, direct proof that rank 2 toric vector bundles on ndimensional projective space split once n is at least 3. This is a result that was originally proved by Bertin and Elencwajg. There is also related work by Kaneyama, Klyachko, and IltenSüss. The idea is that, after possibly twisting the vector bundle, there is a section whose zero locus is a complete intersection.
abstract.
Motivated by the study of rationally connected fibrations (and the MRC quotient) we study different notions of birationally simple fibrations. We say a fibration of smooth projective varieties is Chow constant if pushforward induces an isomorphism on the Chow group of 0cycles. Likewise we say a fibration is cohomologically constant if pullback induces an isomorphism on holomorphic pforms for all p. Our main result is the construction of maximal Chow constant and cohomologically constant fibrations. The paper is largely self contained and we prove a number of basic properties of these fibrations. One application is to the classification of "rationalizations of singularities of cones." We also consider consequences for the Chow groups of the generic fiber of a Chow constant fibration.
abstract.
We show that complex Fano hypersurfaces can have arbitrarily large degrees of irrationality, in particular we give the first examples of Fano varieties with degree of irrationality at least 4. This is difficult because the main technique for bounding the degree of irrationality is to use the existence of holomorphic differential forms. To prove our result we introduce a new specialization method, that is we show that the invariant: ``the minimal degree of a dominant rational map to a ruled variety" can only drop on special fibers in a family. We then reduce modulo a prime p and use an idea of Kollár: differential forms can appear modulo p even when they are not expected in characteristic 0.
abstract.
The purpose of this paper is to compute the degree of irrationality of hypersurfaces of sufficiently high degree in various Fano varieties: quadrics, Grassmannians, products of projective space, cubic threefolds, cubic fourfolds, and complete intersection threefolds of type (2,2). This extends the techniques of Bastianelli, De Poi, Ein, Lazarsfeld, and the second author who computed the degree of irrationality of hypersurfaces of sufficiently high degree in projective space. A theme in the paper is that the fibers of low degree rational maps from the hypersurfaces to projective space tend to lie on curves of low degree contained in the Fano varieties. This allows us to study these maps by studying the geometry of curves in these Fano varieties.
abstract.
abstract.
The purpose of this paper is to study the Zariski tangent space of the punctual Hilbert scheme parametrizing subschemes of a smooth surface which are supported at a single point. We give a lower bound on the dimension of the tangent space in general and show the bound is sharp for subschemes of the affine plane cut out by monomials. Furthermore for monomial subschemes we give an explicit combinatorial formula for the dimension of the tangent space.
abstract.
The purpose of this paper is to explore the geometry and establish the slope stability of tautological vector bundles on Hilbert schemes of points on smooth surfaces. By establishing stability in general we complete a series of results of Schlickewei and Wandel who proved the slope stability of these vector bundles for Hilbert schemes of 2 points or 3 points on K3 or abelian surfaces with Picard group restrictions. In exploring the geometry we show that every sufficiently positive semistable vector bundle on a smooth curve arises as the restriction of a tautological vector bundle on the Hilbert scheme of points on the projective plane. Moreover we show the tautological bundle of the tangent bundle is naturally isomorphic to the sheaf of vector fields tangent to the divisor which consists of nonreduced subschemes.