Pieter Belmans
http://pbelmans.ncag.info/
Thu, 13 Feb 2020 07:39:11 +0000Thu, 13 Feb 2020 07:39:11 +0000Jekyll v3.8.5Fortnightly links (100)<ul>
<li><p><a href="https://ega.fppf.site/">a translation project for Grothendieck's EGA</a> is an English translation of Éléments de géométrie algébrique</a>. There is a <a href="https://fppf.site/ega/book-auto.pdf">beautifully typeset pdf</a>, and the website is using the same framework (hence has all the same features) as the Stacks project. I'm really glad someone has taken Gerby and done something with it!
<li><p><a href="https://schms.math.berkeley.edu/events/miami2020/">Homological Mirror Symmetry and Topological Recursion</a> is the conference webpage for a recent conference in Miami, part of the Simons Collaboration on Homological Mirror Symmetry. And as usual for conferences in this collaboration, there are excellent videos and lecture notes for many talks.
<p>I strongly urge you take a look at the third lecture of Kontsevich. I have stated it before, and I will state it again: the details for those results are something I am really looking forward to!
</ul>
<p><small>Nothing special happened for the 100th fortnightly links. Except it being a few days late, and not containing any arXiv links (did I not pay proper attention?)
Wed, 05 Feb 2020 00:00:00 +0000
http://pbelmans.ncag.info/blog/2020/02/05/fortnightly-links-100/
http://pbelmans.ncag.info/blog/2020/02/05/fortnightly-links-100/fortnightly linksmathematicsSeminar announcement: exceptional collections on partial flag varieties<p>Next semester (which in Bonn runs from roughly April to July) we will be running a (graduate) seminar on exceptional collections, with a view towards partial flag varieties. The two main goals are:
<ul>
<li>to familiarise ourselves with derived categories of finite-dimensional algebras, and important concepts such as Ringel duality and quasi-hereditary algebras
<li>to familiarise ourselves with the geometry of partial flag varieties, and their derived categories, to understand the existence and structure of exceptional collections
</ul>
<p>We will have an organisational meeting this Wednesday, at 10h15 in room 0.011. If you would like to give a talk but can't make it on Wednesday, please send me an email, especially if you are a Bonn graduate student.
<p>For more details (to come), see the <a href="/teaching/exceptional-collections-2020">seminar webpage</a>, with a detailed program likely appearing tomorrow, and otherwise on Wednesday.
Mon, 03 Feb 2020 00:00:00 +0000
http://pbelmans.ncag.info/blog/2020/02/03/graduate-seminar/
http://pbelmans.ncag.info/blog/2020/02/03/graduate-seminar/mathematicsFortnightly links (99)<ul>
<li><p><a href="https://arxiv.org/abs/2001.04148v1">Alexander Kuznetsov, Maxim Smirnov: Residual categories for (co)adjoint Grassmannians in classical types</a> is a cool paper, explaining how the structure of minimal Lefschetz exceptional collections for generalised Grassmannians is reflected by the finer structure in the quantum cohomology of these variaties, and constructs the first (proven to be full) exceptional collection for $\mathrm{OGr}(2,2n)$. Cool stuff!</p>
<li><p><a href="https://arxiv.org/abs/2001.04774v1">Andreas Hochenegger, Ciaran Meachan: Frobenius and spherical codomains and neighbourhoods</a> constructs interesting subcategories of the derived category of a smooth projective variety, associated to natural functors of geometric origin, <em>which are usually not admissible</em>. In my usual setting, all subcategories are automatically admissible (because everything I consider has a Serre functor, usually), but this fails in these examples. This is intriguing me, and I'd love to understand these categories better.
<li><p><a href="https://arxiv.org/abs/2001.05995v1">Max Lieblich, Martin Olsson: Derived categories and birationality</a> studies when a derived equivalence means that varieties are birational. In low dimensions (i.e. up to dimension 2), we know that derived equivalent means isomorphic. But already for Calabi–Yau 3-folds there exist derived equivalent examples which are not birational (which leads to many interesting constructions in the Grothendieck ring of varieties, but that is a different topic). As summarised in question 1.10, the idea is that a derived equivalence which preserves the codimension filtration on cohomological realisations is a way of recognising that the varieties are birational. Interesting stuff!
</ul>
Sun, 19 Jan 2020 00:00:00 +0000
http://pbelmans.ncag.info/blog/2020/01/19/fortnightly-links-99/
http://pbelmans.ncag.info/blog/2020/01/19/fortnightly-links-99/fortnightly linksmathematicsFortnightly links (98)<ul>
<li><p><a href="http://kasmana.people.cofc.edu/MATHFICT/">MathFiction</a> is a collection of literature which involves mathematics or mathematicians in one way or another. Have fun browsing around and looking for good things to read!
<li><p><a href="https://arxiv.org/abs/2001.00536">Weiqiang He, Alexander Polishchuk, Yefeng Shen, Arkady Vaintrob: A Landau-Ginzburg mirror theorem via matrix factorizations</a> proves an <em>all-genus</em> mirror theorem for invertible quasihomogeneous singularities. I'm far from an expert on these matters, but having an all-genus mirror theorem sounds like a very strong result. I'm not aware of any other settings in which this is known (but I am far from an expert). Cool stuff!
<li><p><a href="https://arxiv.org/abs/1912.08970">Benjamin Antieau, Elden Elmanto: Descent for semiorthogonal decompositions</a> shows in complete generality that semiorthogonal decompositions are determined fppf locally. There existed various results of this nature in the literature, but this proves it in all settings. It also works for arbitrary shapes, and I'm looking forward to understanding this in some examples!
<p><small>I must add that we are currently finishing a preprint which proves something of a similar nature, in a more restricted setting giving a stronger description of the resulting fppf stack. Stay tuned for that!
</ul>
Sun, 05 Jan 2020 00:00:00 +0000
http://pbelmans.ncag.info/blog/2020/01/05/fortnightly-links-98/
http://pbelmans.ncag.info/blog/2020/01/05/fortnightly-links-98/fortnightly linksmathematicsFortnightly links (97)<ul>
<li><p><a href="https://arxiv.org/abs/1912.06935">Alexander Perry, Laura Pertusi, Xiaolei Zhao: Stability conditions and moduli spaces for Kuznetsov components of Gushel-Mukai varieties</a> develops the machinery on stability conditions in families in the case of Gushel–Mukai varieties. These are (at least in dimension 4) somewhat parallel to cubic 4-folds, with very precise conjectures about their geometry and rationality. For cubics there are two conjectures concerning their rationality (one in terms of a Hodge-theoretically associated K3 surface, due to Hassett, and one in terms of a categorically associated K3 surface, due to Kuznetsov) which are recently proven to be equivalent.
<p>The intriguing result (for me) is that for Gushel–Mukai 4-folds the analogous conjectures are <em>not</em> equivalent!
<li><p><a href="https://arxiv.org/abs/1912.04957v1">Yves André, Luisa Fiorot: On the canonical, fpqc, and finite topologies on affine schemes. The state of the art</a> discusses how some very classical questions regarding Grothendieck topologies on affine schemes, and relates them to very recent techniques (involving splinters, and prisms). The <a href="https://pbelmans.ncag.info/topologies-comparison/">geographer in me</a> likes these types of results.
<li><p><a href="https://arxiv.org/abs/1912.04332v1">Chunyi Li, Howard Nuer, Paolo Stellari, Xiaolei Zhao: A refined derived Torelli Theorem for Enriques surfaces</a> discusses how for all but a codimension-2 subset of the moduli space of Enriques surfaces, the "Kuznetsov component" of the derived category of the Enriques surface (orthogonal to 10 completely orthogonal line bundles) determines the surface uniquely. I have seen talks about this result, and I quite like the approach of studying 3-spherical objects!
</ul>
Thu, 26 Dec 2019 00:00:00 +0000
http://pbelmans.ncag.info/blog/2019/12/26/fortnightly-links-97/
http://pbelmans.ncag.info/blog/2019/12/26/fortnightly-links-97/fortnightly linksmathematicsGerby supports TikZ now<p>One aspect in which <a href="https://gerby-project.github.io/">Gerby</a> (the software which underlies the Stacks project) was lacking was its support for TikZ and commutative diagrams. Luckily both <a href="https://stacks.math.columbia.edu">The Stacks project</a> and <a href="https://kerodon.net">Kerodon</a> are written by people who use <var>xypic</var> for their commutative diagrams, which allows the use of <a href="https://sonoisa.github.io/xyjax/xyjax.html">XyJax</a>, but this is not a perfect implementation of <var>xypic</var>.
<p>The system on which (part of) Gerby is built is <a href="http://plastex.github.io/plastex/">plasTeX</a>, which contained support for TikZ for a while now, by running <var>pdflatex</var> on every TikZ environment and then converting the resulting pdf to svg. I have now copied over this implementation (thank you <a href="https://www.math.u-psud.fr/~pmassot/en/">Patrick Massot</a> for implementing it for the HTML5 renderer!) and added a <a href="https://github.com/gerby-project/hello-world/blob/master/document-tikz.tex">minimal working example containing TikZ</a>.
<p>The workflow is exactly the same as before, but <strong>I have not tested it</strong> outside the minimal working example. One needs to make sure <var>pdf2svg</var> and <var>pdflatex</var> with the appropriate packages are available.
Sat, 21 Dec 2019 00:00:00 +0000
http://pbelmans.ncag.info/blog/2019/12/21/gerby-supports-tikz/
http://pbelmans.ncag.info/blog/2019/12/21/gerby-supports-tikz/programmingmathematicscomputer sciencele superficie algebriche: an update<img src="/assets/superficie-screenshot.png" width="254" style="float: right">
<p>More than 4 years ago <a href="https://math.commelin.net">Johan Commelin</a> and I created <a href="https://superficie.info">superficie algebriche</a>, an interactive visualisation of the Enriques–Kodaira classification of minimal smooth algebraic surfaces (see the <a href="/blog/2015/07/15/le-superficie-algebriche/">blog post</a>). I'm visiting Johan in Freiburg at the moment, and we've just now
<ul>
<li>added more surfaces (today we've added surfaces with $\mathrm{p}_{\mathrm{g}}=\mathrm{q}=3$ and $\mathrm{p}_{\mathrm{g}}=\mathrm{q}=2$, over the past years we've added various others)
<li>added functionality to add references to MathSciNet and arXiv
</ul>
<p><strong>Wanted</strong> Are you an expert on algebraic surfaces and do you want to see more surfaces added here? Please get in touch then! It is extremely easy to add surfaces, as long as we know where in the literature to find good descriptions for them.
<p>Some ideas of the things we would like to add now that we understand the geometry of algebraic surfaces a bit better than we did in 2015 are:
<ul>
<li>properties of the Albanese map
<li>properties of bicanonical map (and others)
</ul>
<p>We're thinking on how to present this in a convenient way now, and please get in touch with any comments you might have!
Thu, 12 Dec 2019 00:00:00 +0000
http://pbelmans.ncag.info/blog/2019/12/12/update-to-superficie-algebriche/
http://pbelmans.ncag.info/blog/2019/12/12/update-to-superficie-algebriche/algebraic geometryprogrammingmathematicsFortnightly links (96)<ul>
<li><p><a href="https://www.math.ens.fr/~debarre/GMvarieties2019.pdf">Olivier Debarre: Gushel–Mukai varieties</a> is an overview article on Gushel–Mukai varieties, written by one of the leading figures in the study of their geometric, mdular, Hodge-theoretic and derived categorical properties. They form a very interesting class of Fano varieties, with exciting links to hyperkähler varieties. The notes summarise several (long) papers by the author, jointly with Alexander Kuznetsov, and the links to other works, and form a very nice read to get familiar with the state-of-the-art.
<li><p><a href="https://arxiv.org/abs/1912.01689">Špela Špenko, Michel Van den Bergh: Comparing the Kirwan and noncommutative resolutions of quotient varieties</a> shows that the noncommutative crepant resolutions for reductive quotient singularities constructed earlier by them are minimal, in the sense that their derived categories embed fully faithfully in the derived categories of the stacky Kirwan resolutions (which are not minimal). Cool stuff!
<li><p><a href="https://arxiv.org/abs/1912.01538">Andrea Petracci: On deformations of toric Fano varieties</a> is a nice overview article on deformation theory of mildly singular varieties, which then applies the machinery to study deformations of toric Fano varieties. It also determines for all 4319 reflexive Fano polytopes of dimension 3 whether they are smooth, have isolated Gorenstein singularities, ordinary double points, or are not smoothable for various reasons.
</ul>
Sun, 08 Dec 2019 00:00:00 +0000
http://pbelmans.ncag.info/blog/2019/12/08/fortnightly-links-96/
http://pbelmans.ncag.info/blog/2019/12/08/fortnightly-links-96/fortnightly linksmathematicsFortnightly links (95)<ul>
<li><p><a href="https://arxiv.org/abs/1911.07949">Yu-Hsiang Liu: Donaldson-Thomas theory for quantum Fermat quintic threefolds</a> is an interesting preprint I would like to read more carefully. Donaldson–Thomas invariants can be defined for CY3 categories, and a quintic hypersurface in a noncommutative $\mathbb{P}^4$ of a specific type gives rise to such a category which is not of purely geometric origin.
<p>An interesting feature of DT theory is that it should be deformation-invariant. But these CY3 categories are <em>not</em> the deformation of the derived category of a quintic 3-fold. Hence the invariants one obtains are distinct from the ones in the commutative setting, as
<figure class="highlight"><pre><code class="language-sage" data-lang="sage">R.<t> = PowerSeriesRing(ZZ)
chi = -200
print prod([1 / (1 - (-t)^n)^n for n in range(1, 20)])^chi</code></pre></figure>
<p>suggests that in the commutative case series from corollary 6.13 is (if I didn't misinterpret things, please tell me if I did!)
\begin{equation*}
\sum_{n=0}^4\operatorname{DT}^n(Y)t^n
=
1 + 200t + 19500t^2 + 1234000t^3 + 56923950t^4
\end{equation*}
<p>I was hoping to see 2875 here, but maybe this is a different setting?
<li><p><a href="https://arxiv.org/abs/1911.08968">Anton Fonarev: Full exceptional collections on Lagrangian Grassmannians</a> shows that the exceptional collection of expected length in $\mathbf{D}^{\mathrm{b}}(\operatorname{LGr}(n,2n))$ constructed by Kuznetsov–Polishchuk is a <em>full</em> exceptional collection. Here $\operatorname{LGr}(n,2n)$ is the <em>Lagrangian Grassmannian</em>, the isotropic Grassmannian associated to the Dynkin diagram of type $\mathrm{C}_n$ and the maximal parabolic at the "special" vertex. Hence another family bites the dust!
<p>I think a good way to waste my time would be to make an overview table of all the known results, saying which partial flag varieties have
<ul>
<li>a (known) full exceptional collection
<li>a (known) Lefschetz decomposition (which is stronger, and more useful)
<li>a conjectural exceptional collection
</ul>
<p>Let me know if you have any suggestions!
<li><p><a href="https://arxiv.org/abs/1911.08949">Alexander Kuznetsov, Yuri Prokhorov: Rationality of Fano threefolds over non-closed fields</a> discusses a complete picture for (uni)rationality for forms of Fano 3-folds of Picard rank 1. Aside from the obvious geometric interest, an important aspect is the structure of the derived category of such varieties. For instance, they prove that a rationally connected threefold with a semiorthogonal decomposition containing only exceptional objects and derived categories of a curve has an intermediate Jacobian which is <em>isomorphic</em> (and not merely isogenous) to the Jacobian of said curve. Cool!
</ul>
Mon, 25 Nov 2019 00:00:00 +0000
http://pbelmans.ncag.info/blog/2019/11/25/fortnightly-links-95/
http://pbelmans.ncag.info/blog/2019/11/25/fortnightly-links-95/fortnightly linksmathematicsFortnightly links (94)<ul>
<li><p><a href="https://atricolfi.github.io/Intro_Enumerative_Geometry.pdf">Andrea Ricolfi: Introduction to enumerative geometry</a> are a beautiful set of lecture notes (in progress) on enumerative geometry. If the link is broken, it should be possible to find the newest version via Andrea's webpage.
<li><p><a href="https://arxiv.org/abs/1911.01242v1">Igor Burban, Yuriy Drozd: Morita theory for non-commutative noetherian schemes</a> amongst other things proves an noncommutative version of Gabriel's reconstruction theorem: the central subscheme is determined purely in terms of the abelian category of quasicoherent sheaves. Cool stuff!
<li><p><a href="https://arxiv.org/abs/1910.11423v1">Ciro Ciliberto, Mikhail Zaidenberg: Lines, conics, and all that</a> gives a big literature overview of Fano schemes of lines, conics and all that (sic). 481 references!
</ul>
Sun, 10 Nov 2019 00:00:00 +0000
http://pbelmans.ncag.info/blog/2019/11/10/fortnightly-links-94/
http://pbelmans.ncag.info/blog/2019/11/10/fortnightly-links-94/fortnightly linksmathematics