New paper: Varieties of general type with many vanishing plurigenera, and optimal sign and sawtooth inequalities

cat-teethLouis Esser, Terry Tao, Chengxi Wang and I posted a new paper on the arXiv. As the list of authors might suggest, the paper uses a surprising combination of techniques.

We construct smooth projective varieties of general type with the smallest known volumes in high dimensions. These are n-folds with volume roughly 1/en3/2. Among other examples, we construct varieties of general type with many vanishing plurigenera, more than any polynomial function of the dimension.

The examples are resolutions of singularities of general hypersurfaces of degree d with canonical singularities in a weighted projective space P(a0,…,an+1). The volume can be calculated explicitly in terms of the weights ai, and so the problem is to make a good choice of these numbers. In the class of examples we consider, we minimize the volume by reducing to a purely analytic problem about equidistribution on the unit circle.

Namely, consider the sawtooth (or signed fractional part) function, the periodic function on the real line which is the identity on the interval (-1/2,1/2]. For each positive integer m and every probability measure on the real line, at least one of the dilated sawtooth functions g(kx) for k from 1 to m must have small expected value, and we determine the optimal bound in terms of m. We also solve exactly the corresponding optimization problem for the sine function.

Equivalently, we find the optimal inequality of the form ∑k=1m ak sin kx≤ 1 for each positive integer m, in the sense that ∑k=1m am is maximal. The figures show examples of these inequalities, which show striking cancellation among dilated sine or sawtooth functions.

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New paper: Varieties of general type with small volume

Chengxi Wang and I posted a new paper on the arXiv.

By Hacon-McKernan, Takayama, and Tsuji, there is a constant r_n such that for every r at least r_n, the r-canonical map of every n-dimensional variety of general type is birational. We give examples to show that r_n must grow faster than any polynomial in n.

Related to this, we exhibit varieties of several types (Fano, Calabi-Yau, or general type) with small volume in high dimensions. In particular, we construct a mildly singular (klt) n-fold with ample canonical class whose volume is less than 1/2^(2^n). The klt examples should be close to optimal.

All our examples come from weighted projective hypersurfaces. These exhibit a huge range of behavior, and it is not at all clear how to find the best weighted hypersurfaces for these problems. It’s an attractive problem to explore in combinatorial number theory.

For example, Gavin Brown and Alexander Kasprzyk’s computer program shows that the smallest volume for a weighted hypersurface of dimension 2 which is quasi-smooth (hence klt) and has ample canonical class is 2/57035, about 3.5 x 10^{-5}. This occurs for a general hypersurface of degree 316 in the weighted projective space P(158,85,61,11). What is the pattern behind these numbers? Chengxi Wang and I found one pattern and used it to produce examples with small volume in all dimensions. But one can try to do better.

Image: Maru attempting to minimize his volume.

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WAGS Virtual, 23–24 April 2021

The Spring 2021 edition of the Western Algebraic Geometry Symposium (WAGS) will take place virtually on 23–24 April 2021. There will be stimulating talks (short and long), a lively panel, informal office hours with established mathematicians, and social events.

For detailed information, and to register, visit the conference website.

Interested participants (regardless of career stage) can also submit information for virtual introductions that will play as a slideshow during the conference.

Image lifted from @JulietteBruce12.

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This weary three-year-old has seen it all. (Go Bruins!)

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March 31, 2021 · 8:06 pm

WAGS @ Pomona, 13–14 November 2020

This kitten sneaked into Magic Door Books in Pomona last summer
The Fall 2020 edition of the Western Algebraic Geometry Symposium (WAGS) will be held on November 13-14, 2020, via Zoom. You can register here. Pomona College is serving as host. The meeting will include an online poster session; please register by November 6 if you would like to present a poster.

Image: Bookstore cat who works at Magic Door Bookstore IV in downtown Pomona.

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Morse theory on singular spaces

tom-kitten-bursting-his-pair-of-pantsMorse theory is a powerful tool in topology, relating the global properties of a smooth manifold X to the critical points of a smooth function on X. In this note I want to consider the possibility of Morse theory on singular spaces. Some of this dream can be made to work in algebraic geometry, where it helps to analyze the Hilbert scheme of points in new cases.

This is related to my joint work with Marc Hoyois, Joachim Jelisiejew, Denis Nardin, Maria Yakerson on “The Hilbert scheme of infinite affine space.” The connection with Morse theory appears in my paper Torus actions, Morse homology, and the Hilbert scheme of points on affine space.

Morse theory studies a smooth function f on a closed Riemannian manifold X using the gradient flow associated to f. That is, from any point on X, move in the direction in which f is (say) decreasing fastest. In the limit, every point of X is attracted to one of the critical points of f.

From the modern point of view known as “Morse homology,” a central part of the theory is to compactify the space of orbits of the gradient flow. The key point is that any limit of orbits of the gradient flow is a broken trajectory, a chain of orbits that connect a sequence of critical points with decreasing values of f.


This situation has an analog in algebraic geometry. Consider a projective variety X with an action of the multiplicative group T = Gm. (Over the complex numbers, this group can also be called C*.) Then the orbits of T on X are analogous to the gradient flow lines in Morse theory. In particular, for every point x in X, the “downward limit” of its orbit, limt→0(tx), is a T-fixed point of X (analogous to a critical point in Morse theory). Over the complex numbers, this T-action can actually be identified with Morse theory for the “Hamiltonian function” on X.

Just as in Morse theory, I show that every limit of T-orbits on a projective variety X is a broken trajectory, a chain of orbits that connect a sequence of T-fixed points. An interesting point is that this works without assuming that X is smooth. So this is a possible model for Morse theory on singular spaces.

I give an application to the Hilbert scheme of points on affine space. Namely, let Hilbd(An) be the space of 0-dimensional subschemes of degree d in affine n-space. And let Hilbd(An,0) be the (compact) subspace of subschemes supported at the origin in An. I show that, over the complex numbers, these two spaces are homotopy equivalent. (Computing the cohomology of either space is a wide open problem, in general.) The proof uses the algebraic version of Morse theory described here, using the action of T on Hilbd(An) coming from the action on An by scaling. I hope to see more applications: torus actions are everywhere in algebraic geometry.

Image: A chubby Tom Kitten in a broken pair of pants, from The Tale of Tom Kitten, by Beatrix Potter.

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How to get started with the Hilbert scheme

a9d0c6f992d5f50fNow that we can attend seminars all over the world, beginning algebraic geometers may be encountering the Hilbert scheme everywhere. At first glance, however, the idea of the Hilbert scheme is so capacious that it can be hard to grasp.

So, in this post, I want to sketch a path that an interested reader (or student seminar) could follow in beginning to understand the Hilbert scheme. (Some of the links may require your library to have access.) The topic came to mind because of my current joint work with Marc Hoyois, Joachim Jelisiejew, Denis Nardin, and Maria Yakerson, “The Hilbert scheme of infinite affine space.”

First, the general definition: for a projective variety (or scheme) X over a field, the Hilbert scheme of X classifies all closed subschemes of X. The existence of the Hilbert scheme (as a union of projective schemes) reflects a basic feature of algebraic geometry: families of algebraic varieties are parametrized by an algebraic variety (or scheme). But then we have to find ways to analyze the Hilbert scheme in cases of interest.

To start the path of reading, there is an excellent introduction by James McKernan, a 10-page set of MIT lecture notes. It discusses a special case, the “Hilbert scheme of points”, with several examples.

The general construction of Hilbert schemes, due to Grothendieck, is outlined in several places. Perhaps the easiest to read is chapter 1 of Harris and Morrison’s book Moduli of Curves. The most technical step of the proof is not included there (roughly, the fact that there is a uniform bound for the equations of all subschemes of projective space with a given Hilbert polynomial). It might be reasonable for learners to come back to this step later. A standard reference for this step is chapter 14 of Mumford’s book Lectures on Curves on an Algebraic Surface.

Although the Hilbert scheme is hard to understand in full detail, there is a clear — computable! — description of its Zariski tangent space at any point, even where it is singular. Namely, if S is a closed subscheme of a projective scheme X over a field, then the tangent space to Hilb(X) at the point [S] is H^0(S, N_{S/X}), the space of global sections of the normal sheaf. Computing this group in examples is essential for getting to grips with the Hilbert scheme. As a start, you can look in chapter 1 of Harris-Morrison or chapter 1 of Kollár’s book Rational Curves on Algebraic Varieties.

Finally, for many applications, it is important to go one step further and understand the “obstruction space” as well as the tangent space for the Hilbert scheme. This is a great setting in which to learn deformation theory. Roughly, the obstruction space tells you the number of equations needed to define the Hilbert scheme near a given point. There are many possible introductions to deformation theory; let me recommend Sernesi’s book Deformations of Algebraic Schemes. Section 3.2 addresses the Hilbert scheme, with examples and exercises.

There is a vast literature on Hilbert schemes in particular settings, such as the Hilbert scheme of points on a surface. But I hope what I’ve said is enough for you to start exploring.

Image by @kernpanik; license CC BY-NC-SA 4.0.

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WAGON (WAGS Online), 18-19 April 2020

The Western Algebraic Geometry Symposium (WAGS) is going online and worldwide! For more details and to register, see the conference webpage.

Image lifted from @littmath.

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Late-night motives* — MoVid-20: motivic video-conference on 15 April 2020 via Zoom

*The first talk is 1 am Pacific time.

To register, please send an email to with the subject “MoVid-20”.

The schedule in Central European Summer Time (aka time in Germany) is as follows.

9:45-10:00 Conference opening
10:00-11:15 Tom Bachmann
11:15-12:00 coffee break
12:00-13:15 Marc Hoyois
13:15-14:30 lunch break
14:30-15:45 Maria Yakerson
15:45-16:30 coffee break
16:30-17:45 Denis Nardin

Here are the titles and abstracts.

Tom Bachmann: Pullbacks for the Rost-Schmid complex
Let F be a “strictly homotopy invariant” Nisnevich sheaf of abelian groups on the site of smooth varieties over a perfect field k. By work of Morel and Colliot-Thélène–Hoobler–Kahn, the cohomology of F may be computed using a fairly explicit “Rost-Schmid” complex. However, given a morphism f : XY of smooth varieties, it is in general (in particular if f is not flat, e.g. a closed immersion) unclear how to compute the pullback map f *: H*(Y,F) → H*(X,F) in terms of the Rost-Schmid complex. I will explain how to compute the pullback of a cycle with support Z such that f-1(Z) has the expected dimension. Time permitting, I will sketch how this implies the following consequence, obtained in joint work with Maria Yakerson: given a pointed motivic space X, its zeroth P1-stable homotopy sheaf is given by π3P13X)-3.

Marc Hoyois: Milnor excision for motivic spectra
It is a classical result of Weibel that homotopy invariant algebraic K-theory satisfies excision, in the sense that for any ring A and ideal I\subset A, the fiber of KH(A) → KH(A/I) depends only on I as a nonunital ring. In joint work with Elden Elmanto, Ryomei Iwasa, and Shane Kelly, we show that this is true more generally for any cohomology theory represented by a motivic spectrum.

Denis Nardin: A description of the motive of $Hilb(A^\infty)$
The Hilbert scheme of points in infinite affine space is a very complicated algebro-geometric object, whose local structure is extremely rich and hard to describe. In this talk I will show that nevertheless its motive is pure Tate and in fact it coincides with the motive of the Grassmannian. This will allow us to give a simple conceptual description of the motivic algebraic K-theory spectrum. This is joint work with Marc Hoyois, Joachim Jelisiejew, Burt Totaro and Maria Yakerson.

Maria Yakerson: Motivic generalized cohomology theories from framed perspective
All motivic generalized cohomology theories acquire unique structure of so called framed transfers. If one takes framed transfers into account, it turns out that many interesting cohomology theories can be constructed simply as suspension spectra on certain moduli stacks (and their variations). This way important cohomology theories on schemes get new geometric interpretations, and so do canonical maps between different cohomology theories. In the talk we will explain the general formalism of framed transfers and
show how it works for various cohomology theories. This is a summary of joint projects with Tom Bachmann, Elden Elmanto, Marc Hoyois, Joachim Jelisiejew, Adeel Khan, Denis Nardin and Vladimir Sosnilo.

Image: Still from The Third Man (1949, dir. Carol Reed).

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Happy Birthday to this peevish two-year-old

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March 31, 2020 · 9:55 pm