Mackey Functor and his kattenspeelgoed

One year old! Enjoying his present from a famous Dutch math journal.

He also still loves his enemy, Bub — a present from a friendly algebraic visitor.


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March 31, 2019 · 8:40 pm

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.


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

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I support UCLA and the UC Libraries

The University of California Libraries are attempting to negotiate a more acceptable agreement with the giant science publisher Elsevier, as they approach the end of the current contract on 31 December 2018. Elsevier is apparently contacting journal editors at some campuses, attempting to mobilize support for the corporation’s position. I support the UC Libraries and, in particular, will not review papers for any Elsevier journal, as suggested by UCLA’s vice chancellor and provost in a letter to the faculty. The Chronicle of Higher Education (CHE) gives a useful account of the letter and its context. In the CHE article, a member of Elsevier’s library advisory board believes that “faculty members ‘are likely to have a much more nuanced relationship’ with the company” than the deans and librarians who see directly the transfer of money from university to corporation.

I don’t have a nuanced relationship with Elsevier. I have been boycotting Elsevier across the board since about 2000 — no refereeing, no service on editorial boards, no submission of papers — inspired by Rob Kirby’s early analysis of the exorbitant costs of Elsevier journals. Eventually, I signed up to the public Cost of Knowledge boycott. My feelings on this front are as strong as ever.

In particular, I am unenthusiastic about the “publish and read” style of agreement that the UC Libraries are pursuing with Elsevier. My reaction is that they seem to protect the growing wheelbarrow of money that an institution commits to give Elsevier but simply relabel what the money is said to be buying. However, I accept that I likely take a harder line than many fellow faculty members. The UC Libraries serve us all, and they have judged “publish and read” to be the best approach for their many constituencies.

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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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WAGS @ Oregon, 6–7 October 2018

The Fall 2018 meeting of the Western Algebraic Geometry Symposium (WAGS) will take place this weekend at the University of Oregon in Eugene.

Registration is free, and in addition to the usual program of stimulating talks, there will be a poster session.

Speakers are:

Roya Beheshti (Washington in St. Louis)
Laure Flapan (Northeastern)
Eugene Gorsky (UC Davis)
Raman Parimala (Emory)
Mark Shoemaker (Colorado State)
Botong Wang (Wisconsin)

Image from AnnieBananyCreations on Etsy.
Additional cat-duck content.

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How to succeed

My new kitten is a highly successful cat — at least he’s successful against me. To find the silver lining in my defeat, I’ve been cataloging the secrets of his success. They turn out, with minimal translation, to be valid advice for mathematicians or, really, for anyone.

  1. If you’re being dragged away, concentrate on what you can take with you.
  2. Don’t worry for a second about how failing makes you look.
  3. If the direct approach is being guarded, try cozying up to your target gradually.
  4. Always have two projects on the go: if one is being guarded, the other may not be.
  5. Be ready to eat anything.
  6. Keep to a schedule.
  7. Patrol regularly for opportunities.

The most powerful secret is not available to the rest of us.

  1. Be charming in the way only a cat can be.

And some probably aren’t good advice.

  1. Always attack people wiping something with a paper towel.

Photo: Mackey Functor at five months, 6 lbs 3 oz.

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