The direct PDF links are often more updated than the arXiv versions.
The mathematics of Grace Murray Hopper
Notice of the American Mathematical Society 66 (March
2019), no. 3, pp. 330-340.
Grace Murray Hopper (1906-1992) is well known as a pioneering
computer scientist and decorated Naval officer. However, what is
often overlooked in accounts of Hopper's life and work is her
mathematical legacy. For the first
time, using archival material from Yale University's collections,
this article illuminates Hopper's foundational
mathematical training as well as the specific contributions of her
1934 Yale Ph.D. thesis advised by Øystein Ore. Her results
concern necessary conditions for the irreducibility of polynomials with
rational coefficients based on a novel construction, called here the
Newton-Hopper polygon, which is related to the modern
notion of archimedean Newton polygon. The bounds that Hopper obtains
on the distance between the Newton-Hopper polygon of a product of
polynomials and the Minkowski sum of their respective Newton-Hopper
polygons, are explained and clarified.
Unramified Brauer groups of conic bundles threefolds in characteristic two
Alessandro Bigazzi, Christian
Boehning, and Hans-Christian
Graf von Bothmer)
preprint arXiv:1806.02668, submitted (29 pp.)
We establish a formula for computing the unramified Brauer group of tame conic bundle threefolds in characteristic 2. The formula depends on the arrangement and residue double covers of the discriminant components, the latter being governed by Artin-Schreier theory (instead of Kummer theory in characteristic not 2). We use this to give new examples of threefold conic bundles defined over the integers that are not stably rational over the complex numbers.
Universal triviality of the Chow group of 0-cycles and the Brauer group
Alessandro Bigazzi, Christian
Boehning, and Hans-Christian
Graf von Bothmer)
International Mathematics Research Notices, rnz171 (2019)
We prove that a smooth proper universally CH0-trivial variety X over a field k has universally trivial Brauer group. This fills a gap in the literature concerning the p-torsion of the Brauer group when k has characteristic p.
Failure of the local-global principle for isotropy of quadratic forms over rational function fields
preprint arXiv:1709.03707 (7 pp.)
We prove the failure of the local-global principle, with respect to all discrete valuations, for isotropy of quadratic forms over a rational function field of transcendence degree at least 2 over the complex numbers. Our construction involves the generalized Kummer varieties considered by Borcea and Cynk-Hulek.
Period-index bounds for arithmetic threefolds
Benjamin Antieau, Colin Ingalls, Daniel Krashen, Max Lieblich)
Inventiones Mathematicae 26 (2019), no. 2, pp. 301-335.
The standard period-index conjecture for the Brauer group of a field
of transcendence degree 2 over a p-adic field predicts that
the index divides the cube of the period. Using Gabber's theory of
prime-to-l alterations and the deformation theory of twisted
sheaves, we prove that the index divides the fourth power of the
period for every Brauer class whose period is prime to 6p,
giving the first uniform period-index bounds over such fields. For
classes whose period is prime to p (and not necessarily prime
to 6), we prove that the index divides the fifth power of the period.
Conic bundles fourfolds with nontrivial unramified Brauer group
Boehning, Hans-Christian Graf von Bothmer, and Alena Pirutka)
Journal of Algebraic Geometry, 2020
We derive a formula for the unramified Brauer group of a general class
of rationally connected fourfolds birational to conic bundles over
smooth threefolds. We produce new examples of conic bundles over
P3 where this formula applies and which have
nontrivial unramified Brauer group. The construction uses the theory of contact surfaces
and, at least implicitly, matrix factorizations and symmetric
arithmetic Cohen--Macaulay sheaves, as well as the geometry of special
arrangements of rational curves in P2. We also prove the
existence of universally CH0-trivial resolutions for the general
class of conic bundle fourfolds we consider. Using the degeneration
method, we thus produce new families of rationally connected fourfolds
whose very general member is not stably rational.
Azumaya algebras without involution
Uriya First and Ben Williams)
Journal of the European Mathematical Society 21 (2019), no. 3, pp. 897–921.
Generalizing a theorem of Albert, Saltman showed that an Azumaya
algebra A over a ring represents a 2-torsion class in the Brauer group
if and only if there is an algebra A' in the Brauer class of A
admitting an involution of the first kind. Knus, Parimala, and
Srinivas later showed that one can choose A' with deg A' = 2 deg A. We
show that 2 deg A is the lowest degree one can expect in
general. Specifically, we construct an Azumaya algebra A of degree 4
and period 2 such that the degree of any algebra A in the Brauer
class of A admitting an involution is divisible by 8. Separately, we
provide examples of Azumaya algebras of degree 2
admitting symplectic involutions, but no orthogonal involutions. These
stand in contrast to the case of central simple algebras of even
degree over fields, where the presence of a symplectic involution
implies the existence of an orthogonal involution and vice versa.
Stable rationality of quadric and cubic surface bundle fourfolds
Boehning and Alena Pirutka)
Journal of Mathematics 4 (2018), no. 3, pp. 732-760.
Our main result is that a very general
hypersurface X of bidegree (2,3) in P2 x
P3 is not stably rational, which completes the
stable rationality analysis of such bidegree (2,n) fourfold hypersurfaces. Via
projections onto the two factors, X is a cubic surface bundle over
P2 and a conic bundle over P3,
and we analyze the stable rationality problem from both these points
of view. We also show that this case provides another example of a
smooth family of rationally connected fourfolds with rational and
nonrational fibers. Along the way, we introduce new quadric surface bundle fourfolds over P2 with discriminant curve of any even degree at least 8, having nontrivial unramified Brauer group and admitting a universally CH0-trivial resolution.
Some non-special cubic fourfolds
Documenta Mathematica 23 (2018), pp. 637-651.
Ranestad and Voisin have showed, quite surprisingly, that the divisor
in the moduli space of cubic fourfolds consisting of cubics "apolar
to a Veronese surface" is not a Noether-Lefschetz divisor. We give
an independent proof of this by exhibiting an explicit cubic
fourfold X in the divisor and using point counting methods
over finite fields to show X is Noether-Lefschetz general. We
also show that two other divisors considered by Ranestad and Voisin are not Noether-Lefschetz divisors.
Cycles, derived categories, and rationality
in Surveys on Recent Developments in Algebraic Geometry, Proceedings of Symposia in Pure Mathematics, vol. 95, Amer. Math. Soc., Providence, RI, 2017, pp. 199-266.
We survey recent developments in the (stable) rationality problem from
the point of view of unramified cohomology and 0-cycles, as
well as derived categories and semiorthogonal decompositions, and
how these perspectives intertwine and reflect each other. For smooth
projective surfaces, we show how the existence of phantom
subcategories of the derived category can be viewed as a stronger
measure of rationality than the universal triviality of the Chow group
of 0-cycles, equivalently, the existence of a decomposition
of the diagonal.
Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields
Proceedings of the London Mathematical Society
117 (2018), no. 1, pp. 1-64
We study the birational properties of geometrically rational surfaces
from a derived categorical perspective. In particular, we give a
criterion for the rationality of a del Pezzo surface
S over an arbitrary field, namely, that its derived category
decomposes into zero-dimensional components. For deg S ≥ 5, we
construct explicit semiorthogonal decompositions by subcategories of
modules over semisimple algebras arising as endomorphism algebras of
vector bundles and we show how to retrieve information about the index of
S from Brauer classes and Chern classes associated to these vector
Surjectivity of the total Clifford invariant and Brauer dimension
Journal of Algebra 443 (2015), 395-421
This article addresses previous attempts to generalized
Merkurjev's theorem--that every 2-torsion Brauer class is represented
by the Clifford algebra of a quadratic form--to base rings and
schemes other than fields. Parimala, Scharlau, and Sridharan found
smooth complete p-adic curves for which Merkurjev's theorem is
equivalent to the existence of a rational theta characteristic. The
main result of this work is that over a smooth proper p-adic curve
X, replacing the Witt group by the total Grothendieck-Witt group of
line bundle-valued quadratic forms, these obstructions vanish; every
2-torsion Brauer class on X is represented by the Clifford invariant
of some line bundle-valued quadratic form. The key new ingrediant is
the actual construction of the Clifford invariant of a line
bundle-valued quadratic form, when the value line bundle is
Universal unramified cohomology of cubic fourfolds containing a plane
Colliot-Thélène, R. Parimala)
in Brauer Groups and Obstruction Problems: Moduli Spaces and
Arithmetic, Progress in Mathematics, vol. 320, Birkhäuser Basel, 2017,
We prove that a general complex cubic fourfold containing a plane has
constant unramified cohomology in degree three over any field
extension of the complex numbers. The proof uses results on the
unramified cohomology of quadrics due to Kahn, Rost, and Sujatha.
This has bearing on the rationality question for cubic fourfolds.
Along the way, we develop some of the foundations of the universal
triviality of the Chow group of 0-cycles with applications to the
rationality problem in algebraic geometry.
Quadric surface bundles over surfaces
R. Parimala and
Documenta Mathematica, Extra Volume: Alexander S. Merkurjev's Sixtieth Birthday (2015), pp. 31-70.
Let f : T -> S be flat morphism of degree 2 of regular integral
schemes of dimension at most 2 with regular branch divisor D.
We establish a bijection between Azumaya quaternion algebras on
T and quadric surface bundles on S with simple
degeneration along D.
This is a manifestation of the exceptional isomorphism
2A1=D2 degenerating to the exceptional
isomorphism A1=B1=C1. Along the
way, we study the orthogonal group schemes, which are smooth yet
nonreductive, of quadratic forms with simple degeneration.
Failure of the local-global principle for isotropy
of quadratic forms over surfaces
Quadratic Forms and Linear Algebraic Groups, Oberwolfach Reports
10 (2013), issue 2, report 31, pp. 1823-1825.
We construct counterexamples to the local-global principle
for isotropy of quadratic forms over the function field of any surface
over an algebraically closed field of characteristic not 2.
The transcendental lattice of the sextic Fermat surface
Boehning and Hans-Christian Graf von Bothmer)
Mathematical Research Letters 20 (2013), no. 6, pp. 1017-1031.
We prove that the integral polarized Hodge structure on the
transcendental lattice of a sextic Fermat surface is
decomposable. This disproves a conjecture of Kulikov related to a
Hodge theoretic approach to proving the irrationality of the very
general cubic fourfold.
Exceptional collections of line bundles on projective homogeneous varieties
Advances in Mathematics 236 (2013), pp. 111-130.
We construct new examples of exceptional collections of line bundles
on the variety of Borel subgroups of a split semisimple linear
algebraic group G of rank 2 over a field. We exhibit exceptional
collections of the expected length for types A2 and B2=C2 and prove
that no such collection exists for type G2. This settles the question
of the existence of full exceptional collections of line bundles on
projective homogeneous G-varieties for split linear algebraic groups G
of rank at most 2.
Cubic fourfolds containing a plane and a quintic del Pezzo surface
Compositio Algebraic Geometry 1 (2014), issue 2, pp. 181--193.
We isolate a class of smooth rational cubic fourfolds X
containing a plane whose associated quadric surface bundle does not
have a rational section. This is equivalent to the nontriviality of
the Brauer class of the even Clifford algebra over the K3 surface
S of degree 2 associated to X. Specifically, we show
that in the moduli space of cubic fourfolds, the intersection of the
divisors C8 and C14 has five
irreducible components. In the component corresponding to the
existence of a tangent conic, we prove that the general member is both
pfaffian and has nontrivial Brauer class. This answers a question of
B. Hassett. Such cubic fourfolds also provide twisted derived
equivalences between K3 surfaces of degree 2 and 14, hence further
corroboration of Kuznetsov’s derived categorical conjecture on the
rationality of cubic fourfolds.
Fibrations in complete intersections of quadrics, Clifford
algebras, derived categories, and rationality problems
Marcello Bernardara and
Journal de mathématiques pures et appliquées 102 (2014), issue
1, pp. 249--291.
We develop the categorical and algebraic tools of relative
homological projective duality and the Morita-invariance of the even Clifford
algebra under quadric reduction by hyperbolic splitting, to study
semiorthogonal decompositions of the bounded derived category of
fibrations in intersections of quadrics. Together
with input from the theory of quadratic forms, we apply these tools in the case
of fibrations in intersections of two quadrics X -> Y having relative dimension 1, 2, or 3, in which case the
fibers are elliptic curves, Del Pezzo surfaces of degree 4, or Fano
three-folds, respectively. In the latter two cases, if Y is the projective line
over an algebraically closed field of characteristic zero, we relate
rationality questions to categorical representability of X.
Vector bundles of rank 4 and A3=D3
International Mathematical Research Notices 2013 (2013), no. 15, pp. 3450-3476.
This article shows how a version of
"Pascal's rule" for vector bundles gives an explicit isomorphism between
the moduli functors represented by projective homogeneous bundles for
reductive group schemes of type A3 and D3.
This is exploited to show that a vector bundle V of rank
4 has a sub or quotient invertible sheaf if and only if the canonical
symmetric bilinear form on its exterior square has a lagrangian subspace.
Under certain hypotheses on the base scheme,
this is equivalent to the vanishing of the Witt theoretic Euler class
e(V) in W4(X, det(V)).
Remarks on the Milnor conjecture over schemes
Galois-Teichmüller Theory and Arithmetic Geometry (Kyoto, 2010), H. Nakamura, F. Pop, L. Schneps, A. Tamagawa eds.,
Advanced Studies in Pure Mathematics 63, 2012, pp. 1-30
This article mostly consists of a survey of results
concerning the attempts to generalize the Milnor conjectures--both
the K-theoretic and quadratic form aspects--to base rings and
schemes other than fields. Included is a nontrivial example of a
p-adic curve (with bad reduction) that does not satisfy the classical
generalization of the Milnor conjectures for quadratic forms. By
explicitly computing its Grothendieck-Witt groups, Brauer group, and
Clifford invariant map, it is shown how the author's previous results
on the Clifford invariants of line bundle-valued quadratic forms
recover the Milnor conjecture for this curve.
Clifford invariants of line bundle-valued quadratic forms
series 33 (2011), 55 pp.
This article concerns an extension of the Clifford invariant (or 2nd
Stiefel-Whitney class) to similarity classes of line bundle-valued
quadratic forms of even rank and fixed discriminant on a scheme (where
2 is invertible). This invariant resides in degree two étale
cohomology with coefficients in a twisted group scheme of order four
and "interpolates" between the classical Clifford invariant and the
1st Chern class (modulo 2) of the value line bundle. This invariant
is further related to classes in the "involutive" Brauer group
associated to generalized Clifford algebras. In rank at most six,
this invariant is related to the classification of line bundle
Open problems on central simple algebras
(with Eric Brussel,
Skip Garibaldi, and Uzi Vishne)
Transformation Groups 16, Issue 1 (2011), pp. 219-264
A survey of past research on central simple algebras and the Brauer
group of a general field, organized as a list of currently open
problems. The open problems surveyed include those concerning
noncyclic algebras in prime degree, crossed products, generation by
cyclic algebras, period-index, rationality of the center of generic
matrices, essential dimension, involutions (the Springer problem),
surfaces over finite fields (Artin-Tate conjecture), tensor product
decompositions, and the birational geometry of Severi-Brauer varieties
(Amitsur conjecture). Where appropriate, results intended to fill
gaps in the literature are also proved.
Brauer Groups and Obstruction Problems: Moduli Spaces and Arithmetic
(with B. Hassett, A. Várilly-Alvarado, and B. Viray)
Progress in Mathematics, Vol. 320, Birkhäuser Basel, 2017.
Brauer groups and related cohomological invariants have long been important tools for analyzing arithmetic questions on algebraic varieties. Two central examples are the Brauer-Manin obstruction to the existence of rational points and the application of unramified cohomology to rationality questions. Recent advances in derived categories, derived equivalences of varieties, and moduli of twisted sheaves have provided new geometric tools for interpreting elements of the Brauer group. The contributions of this volume explore the implications of these ideas for arithmetic geometry, and hone geometric tools with a view towards future arithmetic applications.
Cohomological invariants of line bundle-valued symmetric bilinear
(submitted version, or the better single spaced version)
Ph.D. thesis at the University of Pennsylvania, May 2009 (advisor Ted Chinburg).
Une démonstration d'un
théorème de Bernstein sur les représentations de
quasi-carré-intégrable de GLn(F)
où F est un corps local non archimédien
(A proof of Bernstein's theorem concerning quasi-square-integrable
representation of GLn(F) where
F is a non-archimedian local field.)
Diplôme d'Études Approfondies (DEA) Mathématiques
Pures thesis at, Université Paris-Sud XI Orsay, France, July
2004 (advisor Guy Henniart).
Volumes of integer polynomials over local fields
(submitted [PDF], with
Undergraduate senior thesis at Reed College, May 2003, (advisor Joe P. Buhler).
Slides/notes from selected talks
Dartmouth Colloquium, Hanover, NH, May 2018
Synapse Resonance, Outreach program of the Yale Scientific Magazine,
New Haven, CT, December 2017
Geometry via point counting
Center for Communications Research (CCR), Princeton, NJ, March 2017
Platonic solids and symmetry
Math Mornings, Yale
University, New Haven, CT, October 2016
Stable rationality of quadric bundles
Geometry over nonclosed fields, Simons Symposium, Schloss Elmau,
Bavaria, Germany, April 2016
Brill-Noether special cubic fourfolds
AMS Summer Research Institute in Algebraic Geometry, Salt Lake City,
Utah, July 2015
Pathways to Science Café, New Haven, CT, October 2014
Quadric surface bundles and quaternion algebras
AMS/MAA Joint Meetings, Special Session on Number Theory and Geometry,
San Diego, CA, January 2013
Azumaya algebras without involution
AMS Special Session on Galois Cohomology and the Brauer Group,
Knoxville, TN, March 2012
Points, lines, and conics
Yale University, Yale Undergraduate Math Society (YUMS), April 2014
Points and conics
University of Connecticut, Math Club, March 2014
Fröhlich twisting via orthogonal motives
AMS Special Session on Hopf Algebras and Galois Module Theory, Tampa,
FL, March 2012
Quaternions and conic sections: from algebra to geometry
Mathematics colloquium, Wake Forest University, Winston-Salem, NC,
Clifford sequences in the theory of line bundle-valued quadratic forms
over arithmetic schemes
AMS Special Session on Arithmetic Geometry, Boca Raton, FL, October 2009
Self-dual Galois representations
Center for Communications Research (CCR) West, La Jolla, CA, November 2008
p-adic polynomial volumes
Graduate student pizza seminar, University of Pennsylvania, October 2004
Which numbers have square roots?
Special thesis talk for trustees, Reed College, Portland, OR, April 2003