Pieter Belmans
http://pbelmans.ncag.info/
Tue, 21 Mar 2023 09:40:17 +0000Tue, 21 Mar 2023 09:40:17 +0000Jekyll v3.9.3Fortnightly links (167)<ul>
<li><p><a href="https://arxiv.org/abs/2302.14499v1">Victoria Hoskins: Moduli spaces and geometric invariant theory: old and new perspectives</a> is a great survey paper on constructions of moduli spaces in algebraic geometry. Highly recommended reading!
<li><p><a href="https://arxiv.org/abs/2303.03436v1">Andreas Hochenegger, Andreas Krug: Asymmetry of $\mathbb{P}$-functors</a> answers a question of Anno and Logvinenko on adjoints of $\mathbb{P}$-functors necessarily being $\mathbb{P}$-functors. Spoiler: the answer is no. The paper is a nifty short argument, and fun to read!
<p><small>What is the plural of Andreas? Andreae?</small>
<li><p><a href="https://arxiv.org/abs/2303.08522v1">Mátyás Domokos: Quiver moduli spaces of a given dimension</a> shows a really cool result in my opinion. Namely that there are only finitely many quiver moduli spaces of a certain dimension. A priori there are only countably many (as the set of quivers, dimension vectors, and chambers in the stability space is countable) but under certain assumptions, it is shown that there are in fact only finitely many.
<p>Thus it is also an interesting question to be able to write down the list of these. In the spirit of <a href="https://arxiv.org/abs/2302.08142">Totaro's question</a> about Bott vanishing for Fano varieties I'd be particularly interested in the Fano case. There is by the way no obvious link between Bott vanishing and Fano quiver moduli: Grassmannians are Fano quiver moduli but fail Bott vanishing, and e.g. <a href="https://www.fanography.info/2-36">2–36</a> is not a Fano quiver moduli space as per the last line of <a href="https://arxiv.org/abs/2001.10556">Fano quiver moduli</a>.
<p>This is closely related to some things I've been thinking about recently, and I hope to be able to share the results soon. Stay tuned!
</ul>
Tue, 21 Mar 2023 00:00:00 +0000
http://pbelmans.ncag.info/blog/2023/03/21/fortnightly-links-167/
http://pbelmans.ncag.info/blog/2023/03/21/fortnightly-links-167/fortnightly linksmathematicsFortnightly links (166)<ul>
<li><p><a href="http://www.math.uni-bonn.de/people/huybrech/CubicPaperAnUpdate.pdf">Daniel Huybrechts: The K3 category of a cubic fourfold – an update</a> is a survey written on the occasion of the Compositio prize which was awarded for the paper <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3705236">The K3 category of a cubic fourfold</a>. Highly recommended if you care about (or wish to be convinced about) understanding cubic fourfolds!
<p>I can also tell you that there is a little work-in-progress coming up real soon, which was directly inspired by this survey article. Stay tuned!
<li><p><a href="https://arxiv.org/abs/2302.08142v1">Burt Totaro: Bott vanishing for Fano 3-folds</a> was already discussed in <a href="/blog/2023/02/16/fortnightly-links-165/">the previous blogpost</a>, but it deserves a spot in the fortnightly links too. Let me in particular advertise the quote
<blockquote>
We can view them as a generalization of toric Fano varieties; they should have some kind of combinatorial classification.
</blockquote>
<p>I look forward to seeing this classification solved!
<li><p><a href="https://arxiv.org/abs/2302.05751">Antonella Grassi, Giulia Gugiatti, Wendelin Lutz, Andrea Petracci: Reflexive polygons and rational elliptic surfaces</a> is an expository and very fun-to-read note on mirror symmetry for del Pezzo surfaces.
</ul>
Mon, 27 Feb 2023 00:00:00 +0000
http://pbelmans.ncag.info/blog/2023/02/27/fortnightly-links-166/
http://pbelmans.ncag.info/blog/2023/02/27/fortnightly-links-166/fortnightly linksmathematicsAn update for Fanography.info: Bott vanishing<p>Last week Burt Totaro uploaded the preprint <a href="https://arxiv.org/abs/2302.08142v1">Bott vanishing for Fano 3-folds</a> to the arXiv, after earlier having studied Bott vanishing for del Pezzo surfaces. The results are now <a href="https://www.fanography.info/bott">Fanography.info</a>. More below the fold.
<hr>
<p>
<p>We say that <strong>Bott vanishing holds</strong> for a smooth projective variety $X$ if $\mathrm{H}^j(X,\Omega_X^i\otimes\mathcal{L})=0$ for all $j\geq 1$, $i\geq 0$ and $\mathcal{L}\in\operatorname{Pic}(X)$ <em>ample</em>. This is a curious property, that we don't understand very well yet:
<ul>
<li>it holds for toric Fano varieties
<li>for a Fano variety it implies $X$ is rigid, thus there are only finitely many Fano varieties in each dimension satisfying Bott vanishing
</ul>
<p>These two properties suggest there might be some combinatorial classification, as alluded to by Burt, but I don't think anyone has a good idea here? More on Bott vanishing:
<ul>
<li>it is expected to fail for every partial flag variety $G/P$ which is not $\mathbb{P}^n$ (see more in <a href="https://arxiv.org/abs/1911.09414">Hochschild cohomology of generalised Grassmannians</a>)
<li>the most curious thing from Totaro's paper is that it is possible that $X$ does not satisfy Bott vanishing, but the blowup of $X$ in some subvariety $Y$ does!
</ul>
<p>Now that your interest is piqued, go take a look at the <a href="https://www.fanography.info/bott">table of all Fano 3-folds satisfying Bott vanishing</a>!
Thu, 23 Feb 2023 00:00:00 +0000
http://pbelmans.ncag.info/blog/2023/02/23/fanography-bott-vanishing/
http://pbelmans.ncag.info/blog/2023/02/23/fanography-bott-vanishing/algebraic geometryprogrammingmathematicsFortnightly links (165)<p>Long overdue, so here goes.
<ul>
<li><p><a href="https://arxiv.org/abs/2301.03149">N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later</a>
is a beautiful retrospective
by the creator of the OEIS, an absolutely amazing resource.
Section 4 is a highly recommended read.
<li><p><a href="https://arxiv.org/abs/2301.04398v1">Wen Chang, Fabian Haiden, Sibylle Schroll: Braid group actions on branched coverings and full exceptional sequences</a>
disproves a conjecture of Bondal–Polishchuk,
which says that the braid group acts transitively on the set of exceptional collections.
Maybe it was an ambitious conjecture, given how little evidence for it there was, but then again,
it might have been a matter of "why wouldn't it be true", in light of the existing evidence.
<p>They use full exceptional collections arising in symplectic geometry,
and not algebraic geometry. So maybe there is something special about exceptional collections arising in algebraic geometry? Who knows!
<li><p><a href="https://arxiv.org/abs/2301.13168v1">Daniel Halpern-Leistner: The noncommutative minimal model program</a>
sets out a precise version of a programme (inspired by mirror symmetry) on the structure of derived categories of varieties.
It is not an easy read, but it is very interesting to see what Dan thinks are the correct precise versions of various mirror symmetry heuristics,
building upon and improving works of others.
I'm looking forward to seeing more details being worked out!
</ul>
Thu, 16 Feb 2023 00:00:00 +0000
http://pbelmans.ncag.info/blog/2023/02/16/fortnightly-links-165/
http://pbelmans.ncag.info/blog/2023/02/16/fortnightly-links-165/fortnightly linksmathematicsMgnbar.info: the geometry of the moduli space of (marked) curves<p>Together with <a href="https://sites.google.com/view/ibarros/">Ignacio Barros</a> we have created <strong><a href="https://mgnbar.info">Mgnbar.info</a></strong>,
a website dedicated to the geometry of $\overline{\mathrm{M}}_{g,n}$, the moduli space of stable marked curves.
<h2>Kodaira dimension of $\overline{\mathrm{M}}_{g,n}$</h2>
<p>Our initial goal was to reproduce a static table that Ignacio had made and which is still available at <a href="https://sites.google.com/view/ibarros/state-of-art-mgn?authuser=0">his homepage</a>, that gives the Kodaira dimension of these moduli spaces.
<p>The reason why this is so interesting is that we've known for more than a hundred years that $\overline{\mathrm{M}}_g$ had Kodaira dimension $-\infty$ (because it is rational) for $g=2,\ldots,10$, by Severi. But then in 1982 Harris and Mumford showed that $\overline{\mathrm{M}}_{25}$ (and $\overline{\mathrm{M}}_g$ for all further odd $g$) are of general type! This was later shown to hold for all $g\geq 24$, and in 2020 for all $g\geq 22$. On the other side we now know that for $g\leq15$ it has Kodaira dimension $-\infty$, and $g=16$ is not of general type.
<p>So there is a drastic change in the complexity of these moduli spaces when you vary $g$. And when you start adding in markings on your curve a similar picture arises, but the change starts happening sooner the more marked points you add.
<h2>What's next?</h2>
<p>Well, that's for you to decide! Right now it's just a showcase of what is possible, with basic interface for Kodaira dimension.
<p>You can make suggestions on <a href="https://github.com/pbelmans/mgnbar/issues">GitHub issues</a>, or send me an email. Please be detailed in what you are suggesting, and include ample references, as I'm no expert.
Wed, 01 Feb 2023 00:00:00 +0000
http://pbelmans.ncag.info/blog/2023/02/01/mgnbar-info/
http://pbelmans.ncag.info/blog/2023/02/01/mgnbar-info/algebraic geometryprogrammingmathematicsFortnightly links (164)<p>We're back with the regular scheduling after having taken the Christmas break off.
<ul>
<li><p><a href="https://twitter.com/stevejtrettel/status/1604865922929287173">Steve Trettel: Barth sextic in $\mathrm{S}^3$</a> is a beautiful visualisation of the Barth sextic, not in an affine patch but rather the double cover ramified in the Barth sextic restricted to $\mathrm{S}^3$ which does manage to capture all of geometry in a single chart.
These pictures are <strong>stunning</strong>!
I wish I could reproduce these and play around with some other surfaces, this is really super cool.
<p>More tweets by Steve Trettel with similar pictures: <a href="https://twitter.com/stevejtrettel/status/1604865922929287173">tweet 1</a>, <a href="https://twitter.com/stevejtrettel/status/1604894528481087488">tweet 2</a>
<li><p><a href="https://arxiv.org/abs/2212.10638v1">James Hotchkiss: Hodge theory of twisted derived categories and the period-index problem</a> and <a href="https://arxiv.org/abs/2212.12971">Aise Johan de Jong, Alexander Perry: The period-index problem and Hodge theory</a> are two preprints discussing the period-index problem using Hodge theory, the former on the categorical side, the latter in a more classical setting. They both look really interesting at first sight, and I'll read them a bit later this week or the next.
<li><p><a href="https://perso.imj-prg.fr/olivier-debarre/wp-content/uploads/sites/34/2022/12/On-rationality-problems-copie.pdf">Olivier Debarre: On rationality problems</a> are lecture notes on rationality questions in algebraic geometry. For a recent update on the state-of-the-art of this question, this is a very nice reference and introduction!
</ul>
Mon, 09 Jan 2023 00:00:00 +0000
http://pbelmans.ncag.info/blog/2023/01/09/fortnightly-links-164/
http://pbelmans.ncag.info/blog/2023/01/09/fortnightly-links-164/fortnightly linksmathematicsFortnightly links (163)<ul>
<li><p><a href="https://arxiv.org/abs/2212.06786v1">Mirko Mauri, Evgeny Shinder: Homological Bondal-Orlov localization conjecture for rational singularities</a> is a cool preprint showing that the essential surjectivity part of the Bondal–Orlov conjecture for resolutions of rational singularities holds, at least if one goes to the Grothendieck group (hence it is true homologically). Nifty!
<li><p><a href="https://pub.math.leidenuniv.nl/~vonkjb/notes/Baskerville.pdf">Jan Vonk: Torsion on elliptic curves</a> is a set of lecture notes for a summer school. It takes the reader on a beautiful trip through the ages, and is a highly recommended read.
<li><p><a href="https://ringtheory.herokuapp.com/">Database of Ring Theory</a> is an interactive repository of properties of rings and modules. Highly fun, and very useful when you're teaching. It's like an interactive version of an algebra version of Counterexamples in Topology.
</ul>
Mon, 19 Dec 2022 00:00:00 +0000
http://pbelmans.ncag.info/blog/2022/12/19/fortnightly-links-163/
http://pbelmans.ncag.info/blog/2022/12/19/fortnightly-links-163/fortnightly linksmathematicsPostdoc position on MathJobs<p><strong><a href="https://www.mathjobs.org/jobs/list/21808">Link to the position</a></strong>. The deadline is <strong>January 20</strong>.
<p><a href="/blog/2022/11/15/postdoc-position/">As explained a bit earlier</a>, I have a postdoc position to work with me. Please spread the news! And if you have any questions, just ask.
Tue, 13 Dec 2022 00:00:00 +0000
http://pbelmans.ncag.info/blog/2022/12/13/postdoc-on-mathjobs/
http://pbelmans.ncag.info/blog/2022/12/13/postdoc-on-mathjobs/mathematicsStacks Project Workshop 2023<p>Great news! There will be another Stacks Project Workshop in 2023, again in Ann Arbor like the 2017 edition.
<p>All information can be <a href="https://stacks.github.io/">found on the website</a>, including the registration, which is open until <strong>February 15, 2023</strong>.
Thu, 08 Dec 2022 00:00:00 +0000
http://pbelmans.ncag.info/blog/2022/12/08/spw-registration/
http://pbelmans.ncag.info/blog/2022/12/08/spw-registration/Stacks projectalgebraic geometryFortnightly links (162)<ul>
<li><p><a href="https://arxiv.org/abs/2211.14195v1">Hans Franzen: A Gelfand-MacPherson correspondence for quiver moduli</a> provides a <em>sweeping</em> generalisation of the fact that there exists an isomorphism of varieties $\operatorname{Gr}(k,n)//\mathbb{G}_{\mathrm{m}}^n\cong(\mathbb{P}^{k-1})^n//\operatorname{SL}_k$ (depending on some choice of stability condition). It says that a moduli space of quiver representations can be written in two ways as the quotient of a certain quiver Grassmannian. Cool!
<li><p><a href="https://arxiv.org/abs/2211.14665v1">Adam Topaz: Algebraic dependence and Milnor K-theory</a> proves that Milnor K-theory fully determines a field (in many situations). I'd love to have a discussion about how this is (not) surprising, for an absolute non-expert it in any case sounds like a wonderful statement.
<li><p><a href="https://arxiv.org/abs/2211.16154v1">Laurent Manivel: A four-dimensional cousin of the Segre cubic</a> talks about generalising the Segre cubic to other (higher-dimensional) settings, and focuses on a 4-dimensional case. Don't forget to read Section 9 for some glimpse into the higher-dimensional cases.
</ul>
Wed, 07 Dec 2022 00:00:00 +0000
http://pbelmans.ncag.info/blog/2022/12/07/fortnightly-links-162/
http://pbelmans.ncag.info/blog/2022/12/07/fortnightly-links-162/fortnightly linksmathematics