Category Archives: math

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.

morse

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 denis.nardin@ur.de 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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WAGS @ Pomona, POSTPONED

This kitten sneaked into Magic Door Books in Pomona last summer

WAGS at Pomona is being postponed until the Fall term.

The Spring 2020 edition of the Western Algebraic Geometry Symposium (WAGS) will be held at Pomona College on 3–4 April 2020. You can register here. Financial assistance is available and can be requested via the registration process.

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

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WAGS @ Utah, 2–3 November 2019

The action-packed Fall 2019 meeting of the Western Algebraic Geometry Symposium (WAGS) will take place at the University of Utah in Salt Lake City.

Personal note: I’ll be attending WAGS and would be interested in chatting with anyone who has experience using MAGMA. — Burt

Speakers are:
Daniel Bragg, Berkeley
Kristin DeVleming, UCSD
Eric Ramos, Oregon
Stefan Schreieder, Munich
Isabel Vogt, Stanford
Jakub Witaszek, Michigan

Registration is free; funding is available for graduate students.

Photo shows a cat sprinting through play between Tigres UANL and Real Salt Lake City at Rio Tinto Stadium in Salt Lake City, 24 July 2019.

Photo credit: Jeff Swinger-USA TODAY Sports.

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SoCalAGS @ USC, 5 October 2019

The fall 2019 meeting of the Southern California Algebraic Geometry Seminar (SoCalAGS) will take place at the University of Southern California (USC) on Saturday 5 October.

The epic slate of speakers is:

Nivedita Bhaskhar (USC)
Stefano Filipazzi (UCLA)
Michail Savvas (UCSD)
Stefan Schreieder (Munich)

Registration is free. Please register, not least because you will get information about … parking.

Image: Very forbearing Trojan Cat who belongs to @groovychickboutique.

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New paper: Bott vanishing for algebraic surfaces

I’ve posted a new paper on the arXiv.

Bott proved a strong vanishing theorem for sheaf cohomology on projective space. It holds for toric varieties, but not for most other varieties. I prove Bott vanishing for the quintic del Pezzo surface, also known as the moduli space M_{0,5}^bar of 5-pointed stable curves of genus zero. This is the first non-toric Fano variety for which Bott vanishing has been shown, answering a question by Achinger, Witaszek, and Zdanowicz.

In another direction, I prove Bott vanishing for many K3 surfaces, including very general K3 surfaces of degree 20 or at least 24. This builds on Beauville and Mukai’s work on moduli spaces of K3 surfaces. It would be interesting to determine exactly which K3 surfaces satisfy Bott vanishing.

In the first draft of the paper, I obtained some results computationally that in the posted version I prove theoretically. Because it’s potentially useful, however, I still give a high-level sketch of a computational approach to proving Bott vanishing.

For people who want more computational detail, I post here an exchange with Nick Addington:

At the end of your paper on Bott vanishing, you talk about choosing a random elliptic K3 of degree up to 38 in Macaulay2. Can you say a little more about how you do it?

Thanks, Nick

Dear Nick,

Sure. I’ll say more about my computations.

First, I have to say more about what I needed to do. Consider a K3 surface with a primitive sublattice Z.{B,f} < Pic(X) (typically we’ll have equality) such that f defines an elliptic fibration pi: X -> P^1 (so f^2 = 0) and B is the ample line bundle I’m interested in. That is, we’ll have B^2 = 20 or B^2 >= 24. The goal is to find one such surface with H^1(X, Omega^1 tensor B) = 0. This will fail for some “special” pairs (X,B), and so we have to make some effort to look among some “general” class of pairs (X,B).

Assume that the elliptic fibration pi has only nodal fibers. Then there are 24 nodes, and you can write down exact sequences that relate the rank-2 bundle Omega^1_X, the rank-1 sheaf Omega^1_{X/P^1}, and the line bundle omega_{X/P^1} on X. You read off from those sequences that H^1(X, Omega^1_X tensor B) = 0 if and only if the 24 nodes impose independent conditions on sections of B+2f. That justifies the decision to consider elliptic K3 surfaces: we have reduced a question about cohomology of a vector bundle to one about sections of a line bundle. (You might need to assume that H^1(X, B-2f) = 0  for this equivalence, but that was easy to check in the cases I considered. Usually B-2f was nef and big, so H^1(X, B-2f) = 0 was immediate from Kawamata-Viehweg vanishing.)

By Riemann-Roch, we know that

h^0(X,B+2f) = ((B+2f)^2+4)/2 = (1/2)B^2 + 2B.f + 2.

Therefore, we cannot hope for the approach above to succeed unless this h^0 is at least 24, which says that:

(*)     (1/2)B^2 + 2B.f – 22 >= 0.

For example, in the hardest case, B^2=20, this inequality says that B.f >= 6. That means that the elliptic K3 surface X -> P^1 is fairly complicated, geometrically: if we use the ample line bundle B to embed the elliptic fibers in projective space, then those curves are elliptic normal curves of degree 6 in P^5.

The easy case is when B^2 >= 40, as Ben Bakker pointed out to me. In this case, the inequality above lets us take B.f = 1. You can use the _same_ elliptic K3 surface to prove Bott vanishing for very general K3s of any degrees at least 40. Just let X -> P^1 be any elliptic K3 surface with section s whose critical locus consists of 24 nodes in distinct fibers (this being known to exist). Then f^2=0, f.s=1, and s^2 = -2. Take B = s+mf for a positive integer m; then B^2 = 2m-2. As long as you take m >= 21 (so B^2 >= 40), you can check by hand that the 24 nodes impose independent conditions on H^0(X, B+2f) = C^{m+3}. (Indeed, you can write down the linear system of B+2f = s+(m+2)f explicitly in this case; it’s just  s  plus pullbacks from H^0(P^1, O(m+2)).)

For B^2 equal to 20 or 24,26,…,38, I looked for the simplest class of K3 surfaces I could think of that came with the desired line bundles B and f (taking B.f to be as small as possible allowed by inequality (*)). For example, for B^2=38 and B.f=2, we have (B-10f)^2 = -2, so we expect B-10f to be effective, and we can look for an embedding of X using the line bundles f and B-10f. (To put it another way: you think about what elements probably generate the Cox ring of X, as a multigraded ring, and use that guess to construct X an a subvariety or covering of a toric variety.)

Sure enough (in this B^2=38 case), this works: take X to be a double cover of the del Pezzo surface W = P(O+O(1)) -> P^1, ramified over a random section of -2K_W. I checked by Macaulay2 that the nodes of the elliptic fibration X -> P^1 impose independent conditions on sections of H^0(X, B+2f). The calculation is made easier because it turns out that you get all those sections by pulling back sections from the appropriate line bundle on the toric surface W.

As B^2 gets lower (while B.f gets bigger as required by inequality (*)), it becomes harder to describe X as a subvariety (or covering) of a toric variety in a way that exhibits all of Pic(X), because the codimension increases. In the hardest case, degree 20, I gave up on that approach and just embedded X in P^5 (so we only see one line bundle on X directly). Here f^2 = 0, B.f = 6, and B^2 = 20, we have (B-f)^2 = 8, and we can use B-f to imbed X as a K3 surface of degree 8 in P^5. I just chose an elliptic normal sextic curve in P^5 to serve as my curve f, and chose a random (2,2,2) complete intersection surface X in P^5 that contains f. It’s easy to describe the elliptic fibration pi: X -> P^1 from this description. Then it was quick for Macaulay2 to compute the critical locus of pi and check that those 24 points impose independent condition on sections of B+2f, as I want. That proves Bott vanishing for very general K3 surfaces of degree 20.

Burt

Image is a puzzle by Hungarian comic artist Gergely Dudás (Dudolf) from the Today show website.

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Complex Algebraic Geometry @ UCSD, 11-13 January 2019

The new year will begin with a Complex Algebraic Geometry conference at UCSD in La Jolla. Registration is free, and there are some funds for graduate students and postdocs.

Speakers are:
Jim Bryan
Paolo Cascini
Paul Hacking
Young-Hoon Kiem
Eric Larson
John Lesieutre
Aaron Pixton
Laura Schaposnik*
Christian Schnell
Jason Starr
Burt Totaro
Chenyang Xu

Image is a still from The Cat Returns (2002, dir. Hiroyuki Morita).

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SoCalAGS @ UCLA, 17 November 2018

11. Sweet Peas and Little black and White cat

The next Southern California Algebraic Geometry Seminar takes place at UCLA on Saturday, 17 November 2018. More information is available at the seminar webpage.

Our excellent slate of speakers is:
Eva Bayer (Lausanne)
Daniel Bragg (Berkeley)
Marc Hoyois (USC)
Junliang Shen (MIT)

Registration is free and very simple. Please register here.

Image of sweetpeas and visiting neighborhood cat from For the Love of Dirt. The 17th of November is sweetpea day (27 Brumaire) in the French Republican Calendar.

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