The direct PDF links are often more updated than the arXiv versions.

Papers/Preprints

  1. Failure of the local-global principle for isotropy of quadratic forms over function fields
    (with V. Suresh)
    accepted in Algebra and Number Theory, arXiv:1709.03707 (15 pp.)
    PDF   arXiv
    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.
  2. Maximal Brill-Noether loci via the gonality stratification
    (with Hannah Larson and Richard Haburcak), preprint arXiv:2310.09954, submitted (12 pp.)
    PDF   arXiv
    We study the restriction of Brill-Noether loci to the gonality stratification of the moduli space of curves of fixed genus. As an application, we give new proofs that Brill-Noether loci of codimension one and two have distinct support, and for fixed rank give lower bounds on when one direction of the non-containments in the Maximal Brill-Noether Loci Conjecture hold. Using these techniques, we also show that Brill-Noether loci corresponding to rank 2 linear systems are maximal as soon as the genus is at least 28 and prove the Maximal Brill-Noether Loci Conjecture in genus 20.
  3. Stickelberger's discriminant theorem for algebras
    (with Owen Biesel and John Voight), American Mathematical Monthly 130 (2023), no. 7, pp. 656-670.
    PDF   arXiv   DOI
    Stickelberger proved that the discriminant of a number field is congruent to 0 or 1 modulo 4. We generalize this to an arbitrary (not necessarily commutative) ring of finite rank over the integers using techniques from linear algebra. Our proof, which relies only on elementary matrix identities, is new even in the classical case.
  4. A census of cubic fourfolds over F2
    (with Avinash Kulkarni, Jack Petok, Jonah Weinbaum), submitted (21 pp.)
    PDF   arXiv
    We compute a complete set of isomorphism classes of cubic fourfolds over F2. Using this, we are able to compile statistics about various invariants of cubic fourfolds, including their counts of points, lines, and planes; all zeta functions of the smooth cubic fourfolds over F2; and their Newton polygons. One particular outcome is the number of smooth cubic fourfolds over F2, which we fit into the asymptotic framework of discriminant complements. Another motivation is the realization problem for zeta functions of K3 surfaces. We present a refinement to the standard method of orbit enumeration that leverages filtrations and gives a significant speedup. In the case of cubic fourfolds, the relevant filtration is determined by Waring representation and the method brings the problem into the computationally tractable range.
  5. Maximal Brill-Noether loci via K3 surfaces
    (with Richard Haburcak), preprint arXiv:2206.04610, submitted (33 pp.)
    PDF   arXiv
    The Brill-Noether loci Mrg,d parameterize curves of genus g admitting a linear system of rank r and degree d$; when the Brill-Noether number is negative, they sit as proper subvarieties of the moduli space of genus g curves. We explain a strategy for distinguishing Brill-Noether loci by studying the lifting of linear systems on curves in polarized K3 surfaces, which motivates a conjecture identifying the maximal Brill--Noether loci. Via an analysis of the stability of Lazarsfeld-Mukai bundles, we obtain new lifting results for line bundles of type g3d which suffice to prove the maximal Brill-Noether loci conjecture in genus 9-19, 22, and 23.
  6. Thinking about abstracts
    Notices of the American Mathematical Society 68, no. 10 (November 2021), pp. 1755-1756.
    DOI
    A column in the November 2021 Early Career section about the history of abstracts in scientific publishing, and how this history can provide some strategies for thinking about writing abstracts.
  7. Explicit descent on elliptic curves and splitting Brauer classes
    (with Benjamin Antieau)
    preprint arXiv:2106.04291, submitted (31 pp.)
    PDF   arXiv
    We prove new results on splitting Brauer classes by genus one curves, settling in particular the case of index 7 classes over global fields. Though our method is cohomological in nature, and proceeds by considering the more difficult problem of splitting μN-gerbes, we use crucial input from the arithmetic of modular curves and explicit N-descent on elliptic curves.
  8. Unramified Brauer groups of conic bundles threefolds in characteristic two
    (with Alessandro Bigazzi, Christian Boehning, and Hans-Christian Graf von Bothmer)
    American Journal of Mathematics 143 (2021), pp. 1601-1631.
    PDF   arXiv   DOI
    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.
  9. Brill-Noether special cubic fourfolds of discriminant 14
    Facets of Algebraic Geometry: A Volume in Honour of William Fulton's 80th Birthday, Volume 1 (editors Paolo Aluffi, David Anderson, Milena Hering, Mircea Mustață, Sam Payne), London Mathematical Society Lecture Note Series 472 (2022), Cambridge University Press, pp. 29-53.
    PDF   arXiv   DOI
    We study the Brill–Noether theory of curves on K3 surfaces that are Hodge theoretically associated to cubic fourfolds of discriminant 14. We prove that any smooth curve in the polarization class has maximal Clifford index and deduce that a cubic fourfold contains disjoint planes if and only if it admits a Brill–Noether special associated K3 surface of degree 14. As an application, we the complement of the pfaffian locus, inside the Noether–Lefschetz divisor of discriminant 14 in the moduli space of cubic fourfolds, is shown to be contained in the irreducible locus of cubic fourfolds containing two disjoint planes.
  10. Conic bundle fourfolds with nontrivial unramified Brauer group
    (with Christian Boehning, Hans-Christian Graf von Bothmer, and Alena Pirutka)
    Journal of Algebraic Geometry 29 (2020), pp. 285-327.
    PDF   arXiv   Macauley2   DOI
    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.
  11. Universal triviality of the Chow group of 0-cycles and the Brauer group
    (with Alessandro Bigazzi, Christian Boehning, and Hans-Christian Graf von Bothmer)
    International Mathematics Research Notices, rnz171 (2019)
    PDF   arXiv   DOI
    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.
  12. The mathematics of Grace Murray Hopper
    Notice of the American Mathematical Society 66 (March 2019), no. 3, pp. 330-340.
    DOI
    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.
  13. Period-index bounds for arithmetic threefolds
    (with Benjamin Antieau, Colin Ingalls, Daniel Krashen, Max Lieblich)
    Inventiones Mathematicae 26 (2019), no. 2, pp. 301-335.
    PDF   arXiv   DOI
    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.
  14. Azumaya algebras without involution
    (with Uriya First and Ben Williams)
    Journal of the European Mathematical Society 21 (2019), no. 3, pp. 897–921.
    PDF   arXiv   DOI
    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.
  15. Stable rationality of quadric and cubic surface bundle fourfolds
    (with Christian Boehning and Alena Pirutka)
    European Journal of Mathematics 4 (2018), no. 3, pp. 732-760.
    PDF   arXiv   DOI
    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.
  16. Some non-special cubic fourfolds
    (with Nick Addington)
    Documenta Mathematica 23 (2018), pp. 637-651.
    PDF   arXiv   DOI
    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.
  17. Cycles, derived categories, and rationality
    (with Marcello Bernardara)
    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.
    PDF   arXiv   BOOK
    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.
  18. Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields
    (with Marcello Bernardara)
    Proceedings of the London Mathematical Society 117 (2018), no. 1, pp. 1-64
    PDF   arXiv   DOI
    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 bundles.
  19. Surjectivity of the total Clifford invariant and Brauer dimension
    Journal of Algebra 443 (2015), 395-421
    PDF   arXiv   DOI
    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 nontrivial.
  20. Universal unramified cohomology of cubic fourfolds containing a plane
    (with Jean-Louis Colliot-Thélène, R. Parimala)
    in Brauer Groups and Obstruction Problems: Moduli Spaces and Arithmetic, Progress in Mathematics, vol. 320, Birkhäuser Basel, 2017, pp. 29-56.
    PDF   arXiv   BOOK
    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.
  21. Quadric surface bundles over surfaces
    (with R. Parimala and V. Suresh)
    Documenta Mathematica, Extra Volume: Alexander S. Merkurjev's Sixtieth Birthday (2015), pp. 31-70.
    PDF   arXiv   DOI
    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.
  22. Cubic fourfolds containing a plane and a quintic del Pezzo surface
    (with Marcello Bernardara, Michele Bolognesi, and Anthony Várilly-Alvarado), Compositio Algebraic Geometry 1 (2014), issue 2, pp. 181--193.
    PDF   arXiv   DOI
    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.
  23. Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems
    (with Marcello Bernardara and Michele Bolognesi)
    Journal de mathématiques pures et appliquées 102 (2014), issue 1, pp. 249--291.
    PDF   arXiv   DOI
    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.
  24. 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.
    PDF   DOI
    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.
  25. The transcendental lattice of the sextic Fermat surface
    (with Christian Boehning and Hans-Christian Graf von Bothmer)
    Mathematical Research Letters 20 (2013), no. 6, pp. 1017-1031.
    PDF   arXiv   DOI   Macauley2
    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.
  26. Exceptional collections of line bundles on projective homogeneous varieties
    (with Alexey Ananyevskiy, Skip Garibaldi, and Kirill Zainoulline)
    Advances in Mathematics 236 (2013), pp. 111-130.
    PDF   arXiv   DOI
    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.
  27. Vector bundles of rank 4 and A3=D3
    International Mathematical Research Notices 2013 (2013), no. 15, pp. 3450-3476.
    [PDF]   arXiv   DOI
    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)).
  28. 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
    PDF   arXiv   BOOK
    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.
  29. Clifford invariants of line bundle-valued quadratic forms
    MPIM preprint series 33 (2011), 55 pp.
    PDF   MPIM
    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 valued-forms.
  30. Open problems on central simple algebras
    (with Eric Brussel, Skip Garibaldi, and Uzi Vishne)
    Transformation Groups 16, Issue 1 (2011), pp. 219-264
    arXiv   DOI
    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.

Books

  1. 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.

Theses

  1. Cohomological invariants of line bundle-valued symmetric bilinear forms
    (submitted version, or the better single spaced version)
    Ph.D. thesis at the University of Pennsylvania, May 2009 (advisor Ted Chinburg).
  2. 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
    (submitted [PDF])
    (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).
  3. Volumes of integer polynomials over local fields
    (submitted [PDF], with hi-resolution images)
    Undergraduate senior thesis at Reed College, May 2003, (advisor Joe P. Buhler).

Notes

Slides from selected talks