Category Archives: math

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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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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Go Canada!

I’m honored to be visiting America’s virtuous neighbor to the north to speak in the PIMSUBC Math Distinguished Colloquium on Friday, September 14, 2018.

Title: Birational geometry and algebraic cycles

Abstract: A fundamental problem of algebraic geometry is to determine which algebraic varieties are rational, that is, isomorphic to projective space after removing lower-dimensional subvarieties from both sides. We discuss the history of the problem. Some dramatic progress in the past 5 years uses a new tool in this context: the Chow group of algebraic cycles.

On the day before, Thursday, September 13, I’ll speak in the UBC Algebraic Geometry seminar.

Title: Hodge theory of classifying stacks

Abstract: The goal of this talk is to create a correspondence between the representation theory of algebraic groups and the topology of Lie groups. The idea is to study the Hodge theory of the classifying stack of a reductive group over a field of characteristic p, the case of characteristic 0 having been studied by Behrend, Bott, Simpson and Teleman. The approach yields new calculations in representation theory, motivated by topology.

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SoCalAGS @ USC, 14 April 2018

The next Southern California Algebraic Geometry Seminar takes place at USC on Saturday, 14 April 2018. More information is available at the seminar webpage.

Our excellent slate of speakers is:
Ben Antieau (UIC)
Barbara Fantechi (Sissa)
Tommaso de Fernex (Utah)
Hiraku Nakajima (Kyoto)

Registration is free and very simple. Please register here.

Photo is by Julie Kitzenberger of a cat and pigeon who are friends. The 14th of April is pigeon day (25 Germinal) in the French Republican calendar.

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Past DLS Lecturers

Every year, the Distinguished Lecture Series (DLS) brings two to four eminent mathematicians to UCLA for a week or more to give a lecture series on their field, and to meet with faculty and graduate students.

2017
Roman Bezrukavnikov
Igor Rodnianski
Claire Voisin
Geordie Williamson
2016
Donald Goldfarb
Benedict Gross
Akshay Venkatesh
2015
Manjul Bhargava
Andrei Okounkov
2014
Robert Bryant
Peter Markowich
Yuval Peres
2013
Michael Aizenman
Ursula Hamenstädt
László Lovász
Gilles Pisier
Richard Taylor
Jean-Pierre Wintenberger
2012
Paul Seidel
2011
Noga Alon
Michael Brenner
Pierre Colmez
Ehud Hrushovski
2010
Michael Harris
Pierre-Louis Lions
Barry Mazur
Ken Ono
Leonid Polterovich
Horng-Tzer Yau
2009
Gregory Margulis
2008
Mario Bonk
John Coates
Elias Stein
Avi Wigderson
2007
Charles Fefferman
C. David Levermore
Shing-Tung Yau
Shouwu Zhang
2006
Peter Sarnak
Peter Schneider
2005
Jean Bellissard
Etienne Ghys
Goro Shimura
Andrei Suslin
Zhengan Weng
2004
Pierre Deligne
Michael Harris
Alexander Lubotzky
2003
Hillel Furstenberg
Robert Langlands
Peter Lax
Nikolai Reshetikhin
Clifford Taubes
Shing-Tung Yau
2002
Raoul Bott
L.H. Eliasson
Dennis Gaitsgory
Jesper Lutzen
Louis Nirenberg
Oded Schramm
I.M. Singer
2001
Michael Atiyah
Jean-Michel Bismut
Alain Connes
Jöran Friberg
David Mumford
Gilles Pisier
Jean-Pierre Serre
Freydoon Shahidi
Gregg Zuckerman
2000
Christophe Deninger
Nessim Sibony
Gang Tian

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