Ostensible contact info:
Actual contact info:
Department of Mathematics
One Shields Avenue
University of California
Davis, CA 95616 USA
|Office hours: M 1:30-2:30, W 3:30-4:30|
|Field: algebraic geometry|
|Research interests: moduli spaces, enumerative geometry, degeneration and deformation techniques, positive-characteristic phenomena.|
My curriculum vitae.
Some of my (older) photos, and
some more recent (mostly bird)
PE 1: Kokikai Aikido Winter 2017
Math 21M: Accelerated Calculus Fall 2016
Math 248B: Algebraic Geometry Winter 2016
Math 21A: Calculus I Fall 2015
Math 248A: Algebraic Geometry Fall 2015
Math 150C: Algebra Spring 2015
Math 115A: Number Theory Fall 2014
Math 248B: Algebraic Geometry Winter 2014
Math 21A: Calculus I Fall 2013
Math 248A: Algebraic Geometry Fall 2013
Math 280: Algebraic Curves and Degenerations Winter 2013
Math 150B: Algebra Winter 2013
Math 21A: Calculus I Fall 2012
Math 248B: Algebraic Geometry Winter 2012
Math 250B: Algebra Winter 2011
Math 16B: Short Calculus Winter 2011
Math 248B: Algebraic Geometry Winter 2010
Math 248A: Algebraic Geometry Fall 2009
Math 21B: Calculus II Fall 2009
Math 240A: Differential Geometry Fall 2008
Math 21C: Calculus III Fall 2008
Math 167: Applied Linear Algebra Summer Session II 2008
Math 250C: Algebra Spring 2008
Number theory reading course Winter 2008
Math 21A: Calculus I Fall 2007
Math 256B: Algebraic Geometry Spring 2007 (at UC Berkeley)
Math 256A: Algebraic Geometry Fall 2006 (at UC Berkeley)
Math H113: Honors Abstract Algebra Spring 2006 (at UC Berkeley)
Math 254A: Number Theory
Fall 2005 (at UC Berkeley)
Western Algebraic Geometry Symposium February 28-March 1, 2015, at UC Davis
Berkeley-Davis-Stanford Algebraic Geometry Colloquium 2006-2011, at UC Berkeley and Stanford (organizer 2006-2007, 2010-2011)
RFG: The geometry and combinatorics of branched covers 2009-2010, at UC Davis
MSRI Summer Graduate Workshop: Deformation Theory and Moduli in Algebraic Geometry July 23-August 3, 2007, at MSRI
Commutative Algebra and Algebraic Geometry Seminar ongoing, at UC Berkeley (organizer 2005-2007)
Western Algebraic Geometry Seminar March 18-19, 2006, in Berkeley
Graduate Number Theory Seminar: Algebraic Number Theory and Elliptic Curves Fall 2000, at MIT
Graduate Number Theory Seminar: Kolyvagin's
Spring 2000, at MIT
Recent progress on vector bundles with sections
3/18/2014, Utah Algebraic Geometry Seminar
An overview of my recent work on higher-rank Brill-Noether theory, including two main branches: modified expected dimensions of moduli spaces on smooth curves, in the case of special determinant; and degeneration techniques. The latter is a mix of foundational work on higher-rank limit linear series, together with applications of this foundational work to proving new existence results on smooth curves.
See also: Brill-Noether loci with fixed determinant in rank 2, Special determinants in higher-rank Brill-Noether theory, Limit linear series moduli stacks in higher rank, and Linked symplectic forms and limit linear series in rank 2 with special determinant (with Montserrat Teixidor i Bigas) below.
New perspectives on limit linear series
9/5/2013, Yale Algebraic and Tropical Geometry Seminar
A seminar talk describing a new way to formulate the definition of Eisenbud-Harris limit linear series, and explaining how this leads to a definition of limit linear series for curves not of compact type.
See also: Limit linear series for curves not of compact type below.
Counting curves on toric surfaces
9/24/2013, Berkeley Commutative Algebra and Algebraic Geometry Seminar
A seminar talk describing joint work with Fu Liu which uses the work of Brugalle and Mikhalkin to show that counts of nodal curves on a certain family of (possibly singular) toric surfaces satisfy universal formulas similar to the Goettsche-Yau-Zaslow formula for smooth surfaces. Also gives combinatorial formulas for the coefficients in the Goettsche-Yau-Zaslow formula.
See also: Severi degrees on toric surfaces (with Fu Liu) below.
Limit linear series: constructions and
November 2010, IMPA lecture series
A series of talks describing the Eisenbud-Harris theory of limit linear series and some applications, and then my own construction of limit linear series spaces, and some new applications of it to fibers of Abel maps and higher-rank Brill Noether theory.
Lifting tropical intersections
12/7/2009, MSRI Tropical Colloquium
Describes joint work with Sam Payne studying when tropicalization commutes with intersection. We describe when this is the case (in a strong sense taking multiplicities into account) in terms of expected dimension hypotheses, considering also the case of intersections inside an ambient subvariety of the torus.
See also: Lifting tropical intersections (with Sam Payne) below.
Vector bundles with sections
10/24/2009, Western Algebraic Geometry Seminar
After reviewing classical Brill-Noether theory, discusses the situation for its higher-rank generalization. Generalizing work of Bertram, Feinberg and Mukai, presents an attempt to study modified expected dimensions systematically in the case of vector bundles of rank 2 with fixed special determinant.
See also: Brill-Noether loci with fixed determinant in rank 2 below.
Beyond Schlessinger: deformation stacks
4/14/2006, Columbia Algebraic Geometry Seminar
Gives quick introductions to classical deformation theory and Schlessinger's criterion, as well as to moduli stacks, and then introduces a new framework for deformation problems, inspired by stack theory, which allows elementary proofs of very general results relating local and global deformations.
See also: deformations and automorphisms: a framework for globalizing local tangent and obstruction spaces below.
The representation theory, geometry, and combinatorics
of branched covers
10/25/2006, Berkeley Representation Theory, Geometry and Combinatorics Seminar
Discusses the general relationship between branched covers of the Riemann sphere and combinatorial group theory, and presents new results on connectedness of some classical Hurwitz spaces (joint with Fu Liu), and on a version of the Riemann existence theorem in characteristic p.
See also: the irreducibility of certain pure-cycle Hurwitz spaces (with Fu Liu), and linear series and existence of branched covers below.
Progress on Riemann existence via
degenerations (or PDF)
4/23/2006, AMS Special Session: Galois Groups in Arithmetic and Geometry, Durham, NH
Discusses two sharp results on existence and non-existence of tame genus-0 covers of the projective line. These are obtained by shifting to the point of view of linear series and using degeneration techniques. Note that Question 11 has since been answered affirmatively, in the irreducibility of certain pure-cycle Hurwitz spaces (with Fu Liu).
See also: linear series and existence of branched covers below.
Limit linear series: an overview
2/25/2006, Bellingham Algebraic Geometry Seminar (joint UBC/UW)
Gives an introduction to the Eisenbud-Harris theory of limit linear series, and proceeds to discuss a functorial compactification, as well as some more recent results on the good properties of this compactification.
See also: a limit linear series moduli scheme, linked Grassmannians and crude limit linear series and flatness of the linked Grassmannian (with David Helm) below.
Rational functions in one variable with given
11/9/2004, Berkeley Commutative Algebra and Algebraic Geometry Seminar
Discusses the number of rational functions on the projective line with prescribed ramification in positive characteristic, giving a formula when the ramification indices are all smaller than p.
See also: rational functions with given ramification in characteristic p below.
Transversality of non-general Schubert
10/23/2004, AMS Special Session: Modern Schubert Calculus, Evanston, IL
Discusses enumerative questions on maps from curves to projective space with prescribed ramification, and a reduction to a question on transversality of Schubert cycles associated to rational normal curves. This question has been answered affirmatively by Mukhin, Varchenko, and Tarasov in this preprint. Touches also on issues of reality.
See also: the number of linear series on curves with given ramification and linear series over real and p-adic fields below.
Elementary applications of a technical theorem
10/18/2004, Brown Algebraic Geometry Seminar
Discusses some applications of a technical theorem of Mochizuki to counting lattice points in certain polytopes, and to studying rational functions with prescribed ramification.
See also: Mochizuki's crys-stable bundles: a lexicon and applications and Mochizuki's indigenous bundles and Ehrhart polynomials (with Fu Liu) below.
Frobenius-unstable vector bundles and the
4/23/2004, Kyoto University Algebraic Geometry Seminar
Surveys results on the Verschiebung rational map induces on moduli spaces of vector bundles on curves by pulling back under Frobenius. Then discusses in more detail the case of genus 2 curves, and vector bundles of rank 2, including a sketch of the degeneration argument for the number of Frobenius-unstable bundles in this case.
See also: the generalized Verschiebung map for curves of genus 2 and Frobenius-unstable bundles and p-curvature below.
Limit linear series and ranks of multiplication maps,
Teixidor i Bigas,
and Naizhen Zhang
(arXiv) Winter 2017
Uses limit linear series on chains of genus-1 curves to study multiplication maps in general, and more specifically to prove an elementary criterion for verifying cases of the Maximal Rank Conjecture. Applies the criterion to give a new proof of the Maximal Rank Conjecture for quadrics, and to prove various other ranges of cases of the conjecture.
Linked determinantal loci and
limit linear series, with John Murray
(arXiv) Fall 2014
Proves that linked determinantal loci are Cohen-Macaulay, implying Cohen-Macaulayness and flatness for limit linear series spaces. Also yields a comparison theorem between two scheme structures on moduli spaces of limit linear series, necessary for analysis of geometry of Brill-Noether loci via degenerations.
Proceedings of the AMS 144 (2016), no. 6, 2399-2410.
Dimension counts for limit linear series on curves
not of compact type (arXiv) Fall 2014
Investigates the dimension behavior of the new definition of limit linear for curves not of compact type, in the context of curves of "pseudocompact type." The first result is that under suitable generality hypotheses, the dimension will be as expected provided that gluing conditions impose the maximal codimension. We then analyze these gluing condition in three specific families, showing that the behavior may depend on chain structures, on enriched structures, or neither. One of these families may be seen as generalizing and shedding new light on the work of Cools, Draisma, Payne and Robeva in the tropical context.
Mathematische Zeitschrift 284 (2016), no. 1-2, 69-93.
Limit linear series for curves
not of compact type (arXiv) Summer 2014
Introduces a new notion of limit linear series for nodal curves not of compact type. Constructs moduli spaces, proves a specialization result, and gives an equivalent definition for a restricted class of curves, showing in particular that the new definition generalizes that of Eisenbud and Harris. Finally, proves a smoothing theorem for the same class of curves, strengthening what was previously known even in the compact type case.
Journal für die reine und angewandte Mathematik (Crelle's journal), to appear.
Limit linear series moduli stacks
in higher rank
(arXiv) Spring 2014
Generalizes the construction of A limit linear series moduli scheme to higher-rank vector bundles and arbitrary curves of compact type, and also gives an equivalent description of Eisenbud-Harris-Teixidor limit linear series, which is new even in the rank-1 case, and yields a proper moduli space in families in that case. Proves a comparison theorem which is the starting point for Linked symplectic forms and limit linear series in rank 2 with special determinant below.
Linked symplectic forms and limit
linear series in rank 2 with special determinant, with
Teixidor i Bigas
(arXiv) Spring 2014
Generalizes the prior linked symplectic Grassmannian construction, applying it to to prove smoothing results for rank-2 limit linear series with fixed special determinant on chains of curves. Applies the general machinery to prove new results on nonemptiness and dimension of rank-2 Brill-Noether loci in a range of degrees.
Advances in Mathematics 288 (2016), 576-630.
Severi degrees on toric surfaces,
(arXiv) Winter 2014
Builds on work of Brugalle and Mikhalkin, Ardila and Block, and Liu to give universal formulas for the number of nodal curves in a linear system on a certain family of (possibly singular) toric surfaces. These formulas are explicitly related to the Goettsche-Yau-Zaslow formula, and are used to give combinatorial expressions for the coefficients arising in the latter.
Journal für die reine und angewandte Mathematik (Crelle's journal), to appear.
Stability of vector bundles on curves and degenerations
(arXiv) Winter 2014
Introduces a weaker and more robust version of stability for vector bundles on reducible curves which is well-suited to degeneration arguments.
Canadian Mathematical Bulletin 59 (2016), no. 4, 858-864.
Relative dimension of morphisms and dimension for algebraic
(arXiv) Spring 2013
Motivated by limit linear series and higher-rank Brill-Noether theory, we introduce a flexible and powerful language for expressing lower bounds on relative dimension of morphisms of schemes, and more generally of algebraic stacks. In a complementary direction, we also develop the basic properties of codimension for algebraic stacks.
Journal of Algebra 437 (2015), 52-78.
Lifting non-proper tropical intersections, with
(arXiv) Summer 2011
For subschemes of the torus with finite intersection, we show that even if the tropicalizations have non-proper intersection, the stable intersection points lift within any given connected component of the tropical intersection, if one works in a suitable partial compactification. The main technical tool is a moving lemma for analytic spaces.
Tropical and Non-Archimedean Geometry, Omid Amini, Matthew Baker and Xander Faber (eds), Contemporary Mathematics 605, 2013, 15-44.
Special determinants in higher-rank Brill-Noether theory
(arXiv) Summer 2011
Continuing the program begun in Brill-Noether loci with fixed determinant in rank 2, we set up a general framework to analyze expected dimensions of higher-rank Brill-Noether loci with fixed special determinant, and prove dimension bounds for some cases for which the dimension of the spaces of global sections is small relative to the rank.
International Journal of Mathematics 24 (2013), no. 11, 1350084, 20 pages.
A simple characteristic-free proof
of the Brill-Noether theorem
(arXiv) Summer 2011
We give a proof of the folklore theorem that limit linear series can be used to give a characteristic-free proof of the Brill-Noether theorem if one uses a different degeneration from that considered by Eisenbud and Harris. The theorem allows ramification at up to two points, and we give a simple, sharp condition for nonemptiness in this setting.
Bulletin of the Brazilian Mathematical Society 45 (2014), no. 4, 807-818 (special issue in honor of Steven Kleiman and Aron Simis).
Abel maps and limit linear series,
(arXiv) Winter 2011
We explore the relationship between fibers of Abel maps and limit linear series for reducible curves consisting of two smooth components glued at a single node. Using the construction of a limit linear series moduli scheme, we associate to a limit linear series a closed subscheme of the fiber of the Abel map, which behaves well under degeneration from a linear series on a family of smooth curves.
Rendiconti del Circolo Matematico di Palermo 62 (2013), no. 1, 79-95 (special issue on algebraic geometry).
Linked alternating forms and linked
symplectic Grassmannians, with
Teixidor i Bigas
(arXiv) Winter 2011
Motivated by applications to higher-rank Brill-Noether theory, we introduce the concepts of linked alternating and linked symplectic forms on a chain of vector bundles, and show that the resulting linked symplectic Grassmannians have good dimensional behavior analogous to that of the classical symplectic Grassmannian.
International Mathematics Research Notices 2014 (2014), no. 3, 720-744.
Strong nonnegativity and sums of
squares on real varieties, with Mohamed Omar (arXiv) Winter 2011
Motivated by scheme theory, introduces a notion of strong nonnegativity for real polynomial functions on varieties. Every sum of squares is strongly nonnegative, so we produce a new obstruction to being a sum of squares. In convex optimization, we recover and generalize obstructions of Gouveia and Netzer to convergence of the theta body hierarchy for certain classes of singular varieties.
Journal of Pure and Applied Algebra 217 (2013), no. 5, 843-850.
Linked Hom spaces
(arXiv) Summer 2010
A note describing a Hom version of linked Grassmannians, and applying it to give a more transparent construction of limit linear series spaces out of linked Grassmannians.
Mathematical Research Letters 18 (2011), no. 2, 329-335.
Lifting tropical intersections,
with Sam Payne
(arXiv) Summer 2010
We use extended tropicalizations and geometry over non-Noetherian valuation rings to prove general results on lifting tropical intersections to algebraic intersections, under an expected dimension hypothesis. Our results apply also to intersections inside ambient subvarieties of the torus.
Documenta Mathematica 18 (2013), 121-175.
Brill-Noether loci with fixed
determinant in rank 2
(arXiv) Spring 2010
Generalizing work of Bertram, Feinberg, and Mukai in the case of fixed canonical determinant, we produce a range of modified expected dimensions for Brill-Noether loci of determinants which are special in suitable senses. Involves a careful study of symplectic Grassmannians and "doubly symplectic Grassmannians."
International Journal of Mathematics 24 (2013), no. 13, 1350099, 24 pages.
Some 4-point Hurwitz numbers in
with Irene Bouw
(arXiv) Summer 2009
We compute a family of Hurwitz numbers in characteristic p, using the theory of stable reduction and formulas of Wewers in combination with degenerations via admissible covers. In the process, we also see that in most of the cases we consider, every cover has a degeneration to a (separable) admissible cover.
Transactions of the AMS 363 (2011), no. 12, 6685-6711.
Functorial reconstruction theorems for stacks,
with Max Lieblich
(arXiv) Summer 2008
We study the following question: to what extent is a stack determined by the associated functor of sets of isomorphism classes of objects? The answer turns out to be that surprisingly often it is completely determined. We show that a number of classical moduli spaces are uniquely determined as algebraic stacks by their functors, including moduli spaces of stable curves with marked points and of stable vector bundles on curves.
Journal of Algebra 322 (2009), no. 10, 3499-3541.
Deformations and automorphisms:
a framework for globalizing local tangent and obstruction spaces
(arXiv) Spring 2008
This paper develops a framework inspired by (but not dependent on) stack theory to treat deformation problems in a uniform but still relatively elementary manner. The immediate goal is to prove in generality comparable to Schlessinger's work certain universal results on global tangent and obstruction spaces, given information on the corresponding local spaces.
Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, IX (2010), no. 3, 581-633.
Linear series and existence of branched covers
(arXiv) Summer 2005, revised
A paper giving new results on existence and, more notably, non-existence of tamely branched covers. Degeneration techniques are used heavily, primarily from the point of view of limit linear series, and a relationship with Mochizuki's indigenous bundles is exploited as well. This has now been heavily revised, with the main result of the irreducibility of certain pure-cycle Hurwitz spaces below allowing for a greatly simplified proof of a stronger version of the main theorem.
Compositio Mathematica 144 (2008), no. 1, 89-106.
The irreducibility of
certain pure-cycle Hurwitz spaces, with
(arXiv) Fall 2006
We use a combination of geometric and group-theoretic techniques to prove that Hurwitz spaces of genus-0 covers of the projective line having a single ramified point over each branch point are irreducible.
American Journal of Mathematics 130 (2008), no. 6, 1687-1708.
and crude limit linear series (arXiv) Spring 2006
A paper comparing the crude limit linear series of a limit linear series moduli scheme below to those of Eisenbud-Harris, showing that the compactification is well-behaved, and giving a new, very direct proof of the Brill-Noether theorem.
International Mathematics Research Notices 2006, no. 25, article ID 25782, 1-27.
Flatness of the linked Grassmannian, with
(arXiv) Spring 2006
A note proving that the linked Grassmannian schemes introduced in a limit linear series moduli scheme are Cohen-Macaulay, reduced, and flat.
Proceedings of the AMS 136 (2008), no. 10, 3383-3390.
Deformations of covers,
Brill-Noether theory, and wild ramification (arXiv) Fall
A note explaining the deformation theory of covers with branching partially specified. Applications are given to positive-characteristic Brill-Noether theory for ramified one-dimensional linear series, with some preliminary investigation into wildly ramified rational functions.
Mathematical Research Letters 12 (2005), no. 4, 483-491.
bundles: a lexicon and applications (arXiv) Fall 2004
A lexicon translating Mochizuki's work into the language of vector bundles with connection, and giving applications to Frobenius-unstable vector bundles and rational functions with prescribed ramification.
Publications of RIMS 43 (2007), no. 1, 95-119.
Verschiebung map for curves of genus 2 (arXiv) Fall 2004
Edited version of chapter IV of my thesis. Discusses the degree and geometry of the Verschiebung map for rank 2 vector bundles on curves of genus 2.
Mathematische Annalen 336 (2006), no. 4, 963-986.
bundles and p-curvature (arXiv) Fall 2004
Edited version of chapters III and VI of my thesis. Uses techniques of connections with vanishing p-curvature to study vector bundles of rank 2 destabilized by Frobenius on curves of genus 2.
Transactions of the AMS 360 (2008), no. 1, 273-305.
with vanishing p-curvature (arXiv) Fall 2004
Edited version of chapter V of my thesis. Relates logarithmic connections with vanishing p-curvature on the projective line to rational maps with prescribed ramification.
Journal of Pure and Applied Algebra 213 (2009), no. 9, 1651-1664.
A limit linear series
moduli scheme (arXiv) Summer 2004
Edited version of chapter II of my thesis. Gives a functorial compactification of the Eisenbud-Harris space of limit linear series.
Annales de l'Institut Fourier, 56 (2006), no. 4, 1165-1205.
Rational functions with
given ramification in characteristic p (arXiv) Summer
Edited version of chapter I of my thesis. Studies the number of rational functions on the projective line with prescribed ramification in positive characteristic, giving a formula when the ramification indices are all smaller than p.
Compositio Mathematica 142 (2006), no. 2, 433-450.
bundles and Ehrhart polynomials, with
(or PDF) Spring 2004
A short paper applying certain finite-flatness results of Mochizuki to obtain identities for numbers of lattice points in different polytopes, and conversely applying the theory of Ehrhart polynomials to show that Mochizuki's indigenous bundles are counted by polynomials in the characteristic of the base field.
Journal of Algebraic Combinatorics 23 (2006), no. 2, 125-136.
over real and p-adic fields
(or PDF) Spring 2004
A short note following up on implications for reality (and p-adicality) of the arguments of the below paper.
Proceedings of the AMS 134 (2006), no. 2, 989-993.
of linear series on curves with given ramification
(or PDF) Spring 2003
A simple paper coming out of, but standing alone from my thesis. Uses the Eisenbud-Harris theory of limit linear series.
International Mathematics Research Notices 2003, no. 47, 2513-2527.
Limit linear series in positive characteristic
and Frobenius-unstable vector bundles on curves
(or PDF) Winter 2004
My PhD thesis, completed at MIT under the direction of Johan de Jong. Also available in default tex formatting (PDF), with a smaller number of pages. These are corrected versions. Finally, there is a list of mathematical errata detailing corrections to the submitted version.
A draft book intended as a substitute for Chapter I of Hartshorne. Includes material on defining abstract varieties via atlases, on complete varieties, and on differential forms.
Limit linear series
A book in progress on the Eisenbud-Harris theory of limit linear series, and its applications. Foundations of the theory are presented in a completely new manner.
Improving the refereeing process: a simple proposal
A suggestion for how to reduce the number of rejections which are both slow and perfunctory.
Notices of the AMS, November 2012.
The Weil conjectures (or
PDF) Winter 2007
A brief exposition of the history of the Weil conjectures and their proof, intended for a very general audience.
In Princeton Companion to Mathematics, Princeton University Press, 2008.
The Riemann Hypothesis for elliptic
curves, with Jasbir Chahal (or
PDF) Fall 2006
An exposition of zeta functions and Riemann hypotheses for global fields, including an elementary proof of the Riemann hypothesis for elliptic curves over finite fields.
American Mathematical Monthly 115 (2008), no. 5, 431-442.
Two degeneration techniques for maps of curves
(or PDF) Spring 2005
Slightly expanded writeup of a talk on admissible covers and limit linear series at the Snowbird Joint Summer Research Conference Algebraic Geometry: Presentations by Young Researchers, in July 2004.
Snowbird Lectures in Algebraic Geometry, 137-143, Contemporary Mathematics 388, AMS, 2005.
Note on cohomology and base change
A rough note collecting general results on the behavior of cohomology sheaves under base change. Primarily intended to help track down references for statements more general than those found in Hartshorne.
Note on dimension theory
A rough note discussing the pathologies of dimension theory even for Noetherian schemes, and ways in which one can phrase dimension-theoretic arguments so that they go through nonetheless.
Connections, curvature, and p-curvature
Rough lecture notes on algebraic connections from both the classical and Grothendieckian point of view, including Mochizuki's definition of p-curvature in the latter setting.