Pieter Belmans
http://pbelmans.ncag.info/
Wed, 20 Sep 2023 06:00:42 +0000Wed, 20 Sep 2023 06:00:42 +0000Jekyll v3.9.3New paper: Hochschild cohomology of Hilbert schemes of points on surfaces<p>I'm really happy and proud to have <a href="https://arxiv.org/abs/2309.06244"><strong>Hochschild cohomology of Hilbert schemes of points on surfaces</strong></a>
on the arXiv now,
joint with my good friends <a href="https://irma.math.unistra.fr/~lfu/">Lie Fu</a>
and <a href="https://sites.google.com/site/andkrugmath/">Andreas Krug</a>.
<p>There is a (to me at least) funny story behind the paper.
In Huybrechts's famous book on Fourier–Mukai transforms,
he briefly discusses Hochschild (co)homology
through a <em>larger</em> and <em>bigraded</em> algebra which
contains both Hochschild cohomology and Hochschild homology,
and also the canonical algebra,
following Orlov's presentation in his survey paper.
It is defined for a smooth projective variety $X$ as
\begin{equation*}
\operatorname{HS}_*^\bullet(X)
=
\bigoplus_{i\in\mathbb{Z}}\bigoplus_{j\in\mathbb{Z}}
\operatorname{HS}_j^i(X)
=
\bigoplus_{i\in\mathbb{Z}}\bigoplus_{j\in\mathbb{Z}}
\operatorname{Ext}_{X\times X}^i(\Delta_*\mathcal{O}_X,\Delta_*\omega_X^{\otimes j}[j\dim X]),
\end{equation*}
where we have (for a good reason)
inserted a certain shift
when compared to the original definition.
To avoid a conflict of terminology,
and emphasise the role of the Serre functor in this definition,
we have decided to call this algebra
the <strong>Hochschild–Serre cohomology</strong> of $X$.
Because this bigraded algebra is a derived invariant,
those 3 invariants are also derived invariants.
<p>But, nowhere had I ever seen other applications of this bigraded algebra.
Was it really useful on its own,
or only as a vehicle for those 3 invariants?
<hr>
<p>
<p>The Hochschild homology of Hilbert schemes of points on surfaces
is easy to compute (as a graded vector space)
because we have Göttsche's formula for the Hodge numbers of Hilbert schemes
(as implemented in the <a href="https://pbelmans.ncag.info/hodge-diamond-cutter/">Hodge diamond cutter</a>, for instance).
Thus, using the Hochschild–Kostant–Rosenberg decomposition,
we can leverage this information to know the dimension of Hochschild homology.
<p>The Hochschild cohomology, however is a different beast.
Sure, we have the Hochschild–Kostant–Rosenberg decomposition,
which involves exterior powers of the tangent bundle now.
But getting a good enough grasp on the tangent bundle of the Hilbert scheme
seems like a challenging problem.
<h3>Two ingredients</h3>
<p>This is where Lie, Andreas, and I tried using the famous
<strong>derived McKay correspondence</strong>
due to Bridgeland–King–Reid–Haiman equivalence,
which gives a derived equivalence
\begin{equation*}
\mathbf{D}^{\mathrm{b}}(\operatorname{Hilb}^nS)
\cong
\mathbf{D}^{\mathrm{b}}([\operatorname{Sym}^nS])
\end{equation*}
where on the right-hand side we have the derived category of
the symmetric quotient stack.
And Hochschild cohomology is a derived invariant,
so we can compute it using the right-hand side.
<p>The second ingredient is the <strong>orbifold Hochschild–Kostant–Rosenberg decomposition</strong>
due to Arinkin, Căldăraru, and Hablicsek.
This can be applied to the right-hand side of the McKay correspondence.
<p>And this is where the magic happens!
When you work out the orbifold decomposition of the Hochschild cohomology,
you suddenly see the Hochschild–Serre cohomology appear.
The reason is that the computation involves fixed loci,
and thus various diagonals,
which when you spell out the details
can be packaged together into one formula which reads
\begin{equation*}
\bigoplus_{n\geq 0}\operatorname{HH}^*(\operatorname{Hilb}^nS)t^n
\cong
\operatorname{Sym}^\bullet(\bigoplus_{i\geq 1}\operatorname{HS}_{1-i}^*(S)t^i)
\end{equation*}
<h3>Conclusion</h3>
<p>Sure, this isn't good enough to write down the Hochschild–Kostant–Rosenberg decomposition
of the Hochschild cohomology of the Hilbert scheme,
but at least we can write down the Hochschild cohomology.
And in fact, for the pieces we care about the most
(namely the first and second Hochschild cohomology)
we know enough to write down the Hochschild–Kostant–Rosenberg decomposition!
<p>In fact, we also:
<ul>
<li>work it all out for arbitrary symmetric quotient stacks
<li>have versions of the result where we allow coefficients
<li>explain various examples from different angles,
and discuss applications to the deformation theory of Hilbert schemes
<li>discuss Hochschild–Serre cohomology
for dg categories,
as a Morita invariant
<li>explain étale functoriality for Hochschild–Serre cohomology
</ul>
so there is plenty of stuff I haven't discussed in this blogpost.
<p>I am really proud of this paper, and I'd be happy to hear all comments,
and answer any questions you might have!
Wed, 13 Sep 2023 00:00:00 +0000
http://pbelmans.ncag.info/blog/2023/09/13/new-paper-hochschild-hilbert/
http://pbelmans.ncag.info/blog/2023/09/13/new-paper-hochschild-hilbert/algebraic geometrymathematicsFortnightly links (172)<ul>
<li><p><a href="https://arxiv.org/abs/2308.06191v1">Ben Webster, Philsang Yoo: 3-dimensional mirror symmetry</a> is an extended version
of a Notices of AMS article which isn't out yet (I think),
in which the authors explain 3-dimensional mirror symmetry,
which is also known as <em>symplectic duality</em>.
I remember being confused several years ago about the relationship between 2d and 3d mirror symmetry,
and whilst that confusion has been cleared up since then,
this article was still a really interesting read!
<li><p><a href="https://thosgood.com/sga/">A translation project for SGA</a> is what it says on the nose.
It uses <a href="https://bookdown.org">Bookdown</a>,
so it is not written in LaTeX,
but rather some Markdown + LaTeX hybrid. Cool!
<li><p><a href="https://arxiv.org/abs/2308.08121">Hiromu Tanaka: Fano threefolds in positive characteristic I</a>,
<a href="https://arxiv.org/abs/2308.08122">II</a>,
<a href="https://arxiv.org/abs/2308.08124">III</a> (with Masaya Asai),
<a href="https://arxiv.org/abs/2308.08127">IV</a>
is a sequence of papers covering <em>many</em> details of the classification of Fano 3-folds in positive characteristic.
I will try to figure out what needs to be changed on <a href="https://fanography.info">Fanography</a>
in order for it to cover the positive characteristic case too.
</ul>
Mon, 04 Sep 2023 00:00:00 +0000
http://pbelmans.ncag.info/blog/2023/09/04/fortnighty-links-172/
http://pbelmans.ncag.info/blog/2023/09/04/fortnighty-links-172/fortnightly linksmathematicsGitHub LaTeX template<p>For the impatient and tech-savvy: go and check out <strong><a href="https://github.com/pbelmans/latex-template">this GitHub template repository</a></strong>.
<p>Last week during the Stacks project workshop I gave an <a href="https://stacks.github.io/2023-git.pdf">introduction to Git for working mathematicians</a>,
and one thing that people found interesting then was my GitHub LaTeX template,
which includes:
<ul>
<li>a GitHub Action to build pdf's upon push, see <a href="https://github.com/pbelmans/latex-template/blob/main/.github/workflows/pdf.yml">pdf.yml</a>
<li>hooks to add commit metadata to pdf's
</ul>
<p>The first thing is the coolest I think,
and a variation upon existing such actions that are available on GitHub already.
It guarantees that there is always a copy of the pdf in the repository,
except that it lives in an orphan branch where it cannot create conflicts!
<p>I've now put this on GitHub as a <strong><a href="https://github.com/pbelmans/latex-template">template repository</a></strong>,
so that you can just click the green <strong>Use this template</strong> button
to start a new repository from scratch, with this configuration.
<hr>
<p>
<p>Is there something missing, or unclear? Let me know!
<p>I also have some inspiration to write a "<em>Git and GitHub for working mathematicians</em>" tutorial,
there exist excellent introductions already,
but I think there needs to be one which doesn't mention the commandline <em>at all</em>.
Stay tuned for that!
Thu, 17 Aug 2023 00:00:00 +0000
http://pbelmans.ncag.info/blog/2023/08/17/github-latex-template/
http://pbelmans.ncag.info/blog/2023/08/17/github-latex-template/mathematicsStatistics for the Stacks project (2)<p>For the Stacks project workshop last week
I crunched the numbers again
<a href="/blog/2020/04/09/view-stats-stacks-project/">like I did back in 2020 to come up with some statistics</a>.
Here's the updated graph.
<p>What is the conclusion?
That the number of algebraic geometers seems to grow linearly over the years?
A scary thought!
<script src="https://d3js.org/d3.v4.js"></script>
<div style="text-align: center" id="unique"></div>
<div style="display: none" id="visits"></div>
<div style="display: none" id="pages"></div>
<script>
// set the dimensions and margins of the graph
var margin = {top: 10, right: 30, bottom: 30, left: 60},
width = 460 - margin.left - margin.right,
height = 400 - margin.top - margin.bottom;
// append the svg object to the body of the page
var unique = d3.select("#unique")
.append("svg")
.attr("width", width + margin.left + margin.right)
.attr("height", height + margin.top + margin.bottom)
.append("g")
.attr("transform",
"translate(" + margin.left + "," + margin.top + ")");
var visits = d3.select("#visits")
.append("svg")
.attr("width", width + margin.left + margin.right)
.attr("height", height + margin.top + margin.bottom)
.append("g")
.attr("transform",
"translate(" + margin.left + "," + margin.top + ")");
var pages = d3.select("#pages")
.append("svg")
.attr("width", width + margin.left + margin.right)
.attr("height", height + margin.top + margin.bottom)
.append("g")
.attr("transform",
"translate(" + margin.left + "," + margin.top + ")");
//Read the data
d3.csv("/assets/stacks-data-2023.csv",
// When reading the csv, I must format variables:
function(d){
return { date : d3.timeParse("%Y-%m")(d.month), unique : d.visitors, visits : d.visits, pages : d.pages }
},
// Now I can use this dataset:
function(data) {
// x axis
var x = d3.scaleTime()
.domain(d3.extent(data, function(d) { return d.date; }))
.range([ 0, width ]);
unique.append("g")
.attr("transform", "translate(0," + height + ")")
.call(d3.axisBottom(x));
visits.append("g")
.attr("transform", "translate(0," + height + ")")
.call(d3.axisBottom(x));
pages.append("g")
.attr("transform", "translate(0," + height + ")")
.call(d3.axisBottom(x));
// y axes
var uniqueY = d3.scaleLinear()
.domain([0, d3.max(data, function(d) { return +d.unique; })])
.range([ height, 0 ]);
unique.append("g")
.call(d3.axisLeft(uniqueY));
var visitsY = d3.scaleLinear()
.domain([0, d3.max(data, function(d) { return +d.visits; })])
.range([ height, 0 ]);
visits.append("g")
.call(d3.axisLeft(visitsY));
var pagesY = d3.scaleLinear()
.domain([0, d3.max(data, function(d) { return +d.pages; })])
.range([ height, 0 ]);
pages.append("g")
.call(d3.axisLeft(pagesY));
// add the lines
unique.append("path")
.datum(data)
.attr("fill", "none")
.attr("stroke", "steelblue")
.attr("stroke-width", 1.5)
.attr("d", d3.line()
.x(function(d) { return x(d.date) })
.y(function(d) { return uniqueY(d.unique) })
)
visits.append("path")
.datum(data)
.attr("fill", "none")
.attr("stroke", "steelblue")
.attr("stroke-width", 1.5)
.attr("d", d3.line()
.x(function(d) { return x(d.date) })
.y(function(d) { return visitsY(d.visits) })
)
pages.append("path")
.datum(data)
.attr("fill", "none")
.attr("stroke", "steelblue")
.attr("stroke-width", 1.5)
.attr("d", d3.line()
.x(function(d) { return x(d.date) })
.y(function(d) { return pagesY(d.pages) })
)
// add axis labels
unique.append("text")
.attr("transform", "rotate(-90)")
.attr("y", 0 - margin.left)
.attr("x",0 - (height / 2))
.attr("dy", "1em")
.style("text-anchor", "middle")
.text("Number of unique visitors per month");
visits.append("text")
.attr("transform", "rotate(-90)")
.attr("y", 0 - margin.left)
.attr("x",0 - (height / 2))
.attr("dy", "1em")
.style("text-anchor", "middle")
.text("Number of visits");
pages.append("text")
.attr("transform", "rotate(-90)")
.attr("y", 0 - margin.left)
.attr("x",0 - (height / 2))
.attr("dy", "1em")
.style("text-anchor", "middle")
.text("Number of pageviews");
})
</script>
Wed, 16 Aug 2023 00:00:00 +0000
http://pbelmans.ncag.info/blog/2023/08/16/new-stats-stacks-project/
http://pbelmans.ncag.info/blog/2023/08/16/new-stats-stacks-project/Stacks projectalgebraic geometryFortnightly links (171)<ul>
<li><p><a href="https://arxiv.org/abs/2307.03174v1">Dan Rogalski: Artin-Schelter regular algebras</a>
is a survey on AS-regular algebras,
the algebras responsible in noncommutative algebraic geometry
for noncommutative projective space.
A must-read for anyone new to the field.
<li><p><a href="https://arxiv.org/abs/2307.13555v1">Hiroshi Iritani: Quantum cohomology of blowups</a>
and
<a href="https://arxiv.org/abs/2307.03696v2">Hiroshi Iritani, Yuki Koto: Quantum cohomology of projective bundles</a>
show that the quantum cohomology of a blowup resp. projective bundle
behave as one would expect,
under the mirror symmetry correspondence
and the fact that we know Orlov's blowup formula and projective bundle formula.
Really cool!
<li><p><a href="http://miurror.stars.ne.jp/calabiyau3data/hodges.html">Hodge numbers for smooth Calabi–Yau 3-folds</a>
and <a href="http://miurror.stars.ne.jp/calabiyau3data/primitiveCY3.html">Topological invariants for known smooth Calabi–Yau 3-folds of Picard number one</a>
are two visualisations of numerical data appearing in the classification of Calabi-Yau 3-folds.
Really <em>really</em> cool!
</ul>
Sun, 06 Aug 2023 00:00:00 +0000
http://pbelmans.ncag.info/blog/2023/08/06/fortnightly-links-171/
http://pbelmans.ncag.info/blog/2023/08/06/fortnightly-links-171/fortnightly linksmathematicsThe identification of a 6-dimensional Kronecker quiver moduli with a certain zero section<p>In the post <a href="/blog/2023/07/06/new-paper-chow-rings-quiver-moduli/">'New paper: On Chow rings of quiver moduli'</a>
I explained a conjectural identification between two varieties, for which we used our newly gained understanding of Chow rings to give more evidence.
<p>Shortly after posting it to the arXiv I got an email by Laurent Manivel,
pointing out that the zero section of $\operatorname{Gr}(2,8)$ makes an appearance
<a href="https://doi.org/10.1515/crll.2005.2005.585.93">Atanas Iliev and Laurent Manivel: Severi varieties and their varieties of reductions</a>,
and that it might be useful to prove the identification.
<p>And indeed, after reading up a bit, I realised that the proof of the conjecture can be given by
simply stringing together various results,
avoiding any actual computations.
Namely:
<ol>
<li>The Kronecker quiver moduli space $X$
is also the moduli space of bundles $Z$ on $\mathbb{P}^2$
with $(r,\mathrm{c}_1,\mathrm{c}_2)=(4,1,3)$,
which is of height zero in the sense of Drezet.
<li>The moduli space $Z$
is the image of the second contraction of $\operatorname{Hilb}^3\mathbb{P}^2$
(the other one being $\operatorname{Sym}^3\mathbb{P}^2$).
<li>The zero section $Y$ is the variety $Y_2$ in Iliev–Manivel,
this might be clear from Iliev–Manivel itself
but a detailed proof is given in Theorem 3.8 of
<a href="https://doi.org/10.5802/aif.3131">Pietro De Poi, Daniele Faenzi, Emilia Mezzetti and Kristian Ranestad: Fano congruences of index 3 and alternating 3-forms</a>.
<li>Finally, $Y$ and $Z$ are identified by Theorem 4.2 in Iliev–Manivel.
</ol>
<p>We will update the paper accordingly, but as Hans and I will be on (separate) vacations for the next few weeks,
I wanted to write this blog post
so that no-one is wasting time on a proof
if one is so readily available from the literature already.
Of course, if you have something else of interest to tell us, please do!
<p>I would like to thank Laurent Manivel for the suggestion, and Jieao Song and Fabian Reede for related discussions.
Fri, 14 Jul 2023 00:00:00 +0000
http://pbelmans.ncag.info/blog/2023/07/14/identification-of-kronecker-moduli/
http://pbelmans.ncag.info/blog/2023/07/14/identification-of-kronecker-moduli/algebraic geometrymathematicsGitHub Actions for automated unit tests of the Hodge diamond cutter<p style="background-color: rgb(255, 243, 205); border: 1px solid rgb(255, 238, 186); padding: 10px;">
The functionality outlined below, and much more, is implemented in <a href="https://github.com/pbelmans/hodge-diamond-cutter"><strong>Hodge diamond cutter</strong></a>, which can be used in Sage.
If you use it for your research, please cite it using <a href="https://doi.org/10.5281/zenodo.3893509" rel="nofollow"><img src="https://zenodo.org/badge/DOI/10.5281/zenodo.3893509.svg" alt="DOI" data-canonical-src="https://zenodo.org/badge/DOI/10.5281/zenodo.3893509.svg" style="max-width:100%;"></a>.
</p>
<p>Today I was supposed to take the day off. So what did I do for the past hour? I set up <a href="https://github.com/features/actions">GitHub Actions</a> to automatically run the <a href="/blog/2021/08/05/hodge-diamond-cutter-documentation/">unit tests</a> that are in the documentation. Yes, this is <em>nerdy</em>.
<p>Why is this interesting? Well, for one it now displays this badge
<p style="text-align: center"><img src="https://github.com/pbelmans/hodge-diamond-cutter/actions/workflows/tests.yml/badge.svg">
<p>on the <a href="https://github.com/pbelmans/hodge-diamond-cutter/tree/master">repository page</a>.
This should say <strong>passing</strong>.
If it doesn't, please tell me (although I probably already know, because I should get an email from GitHub if the test fails).
<p>Whenever something happens with the code (a commit, or a pull request, or manually triggering the tests),
<a href="https://github.com/pbelmans/hodge-diamond-cutter/blob/master/.github/workflows/tests.yml">this script</a>
will
<ul>
<li>start up an Ubuntu virtual machine
<li>use the SageMath Docker container
<li>install the Hodge diamond cutter
<li>run the tests (of which there were 251 at the time of writing)
</ul>
<p>Here is the <a href="https://pipelines.actions.githubusercontent.com/serviceHosts/6805fe20-7399-43c4-860b-1f473ca3f60b/_apis/pipelines/1/runs/19/signedlogcontent/2?urlExpires=2023-07-07T14%3A35%3A19.8345770Z&urlSigningMethod=HMACV1&urlSignature=J3JIeS296pIjP3W1AdpOAwFuzkeZB0Bhval0nr%2Fuq0w%3D">log file</a> of the last test run today. The important part is
<code><pre>2023-07-07T14:24:02.8409203Z [251 tests, 3.58 s]
2023-07-07T14:24:02.8409755Z ----------------------------------------------------------------------
2023-07-07T14:24:02.8410066Z All tests passed!
2023-07-07T14:24:02.8410405Z ----------------------------------------------------------------------
2023-07-07T14:24:02.8410691Z Total time for all tests: 3.7 seconds
2023-07-07T14:24:02.8412365Z cpu time: 3.5 seconds
2023-07-07T14:24:02.8412605Z cumulative wall time: 3.6 seconds</pre></code>
<p>telling us that all went well.
<p>I believe the workflow is really easy to use yourself (after I spent too long getting it to work, as simple as it is), and I hope someone else who develops things for SageMath is interested in using a similar test workflow.
<h3>Installation instructions</h3>
<p>I also restructured the repository, so that installing it through pip is now easy.
Just follow <a href="https://github.com/pbelmans/hodge-diamond-cutter#getting-started">the instructions in the documentation</a>.
Fri, 07 Jul 2023 00:00:00 +0000
http://pbelmans.ncag.info/blog/2023/07/07/hodge-diamond-cutter-github-actions/
http://pbelmans.ncag.info/blog/2023/07/07/hodge-diamond-cutter-github-actions/algebraic geometrymathematicsNew paper: On Chow rings of quiver moduli<p>I'm really happy to have <strong><a href="https://arxiv.org/abs/2307.01711">On Chow rings of quiver moduli</a></strong> on the arXiv now. It is joint with my good friend <a href="https://math.uni-paderborn.de/ag/arbeitsgruppe-algebra/team/hans-franzen">Hans Franzen</a>, with whom I also wrote (together with others) the topic of the <a href="/blog/2022/10/04/new-paper-no-git-quiver-git/">previous "new paper" post</a>.
<p>The motivation is somewhat similar as to the previous post: there are lots of interesting parallels between
<ul>
<li>moduli of semistable quiver representations (or quiver moduli, in short)
<li>moduli of semistable vector bundles on curves
</ul>
<p>As the title suggests, we now care about Chow rings (and cohomology rings) of these moduli spaces.
First of all, we'd like an explicit presentation.
For moduli of vector bundles on curves, a nice introduction can be found in <a href="https://mathscinet.ams.org/mathscinet/article?mr=1489212">Alastair King, Peter Newstead: On the cohomology ring of the moduli space of rank 2 vector bundles on a curve</a>.
<p>For quiver moduli, this was done in <a href="https://mathscinet.ams.org/mathscinet/article?mr=3318266">Hans Franzen: Chow rings of fine quiver moduli are tautologically presented</a>, building upon <a href="https://mathscinet.ams.org/mathscinet/article?mr=1324213">Alastair King and Charles Walter: On Chow rings of fine moduli spaces of modules</a>.
<h3>Todd classes and point classes</h3>
<p>But to do calculations, as for instance implemented in the wonderful <a href="http://www2.macaulay2.com/Macaulay2/doc/Macaulay2-1.20/share/doc/Macaulay2/Schubert2/html/index.html">Schubert2</a> for partial flag varieties (in type A), one also needs
<ul>
<li>an expression for the point class (so that you know how you are integrating things)
<li>an expression for the Todd class, to make Hirzebruch–Riemann–Roch calculations possible
</ul>
<p>in terms of the given presentation. I don't know where this would be done for moduli of vector bundles on curves. But for quiver moduli it is now done in our new paper.
<p>The trick is to upgrade the usual 4-term sequence of Hom and Ext of quiver representations
to a <strong>4-term sequence of vector bundles on the base of a family</strong>.
This is not new, but I am really happy with how we approached this problem,
as I think it highlights how quiver moduli are parallel to more conventional moduli spaces of sheaves.
<p>Another important ingredient, again looking for parallels with more conventional moduli spaces of sheaves,
is the <strong>Kodaira–Spencer morphism</strong>.
Its construction was sketched by Dyer and Polishchuk in <a href="https://mathscinet.ams.org/mathscinet/article?mr=3925499">NC-smooth algebroid thickenings for families of vector bundles and quiver representations</a>,
but we give a novel take on it, explaining the role of Atiyah classes.
<h3>A conjecture</h3>
<p>Our initial motivation was to do some explicit calculations in a specific case.
Namely, Kronecker moduli, or Kronecker quiver moduli,
are quiver moduli associated to the easiest possible quivers: Kronecker quivers,
with 2 vertices and $n$ arrows.
To make things interesting we assume $n\geq 3$.
<p>In many cases these Kronecker quiver moduli have an interpretation as a Grassmannian.
But not always.
The smallest case
is the 3-Kronecker quiver, with dimension vector $(2,3)$.
It is a 6-dimensional Fano variety of Picard rank 1 and index 3,
with Hodge numbers $\mathrm{h}^{p,p}$ given by
\[
1, 1, 3, 3, 3, 1, 1.
\]
<p>There is a great interest in describing Fano varieties
in terms of zero loci of equivariant vector bundles on partial flag varieties.
When prompted with this challenge,
<a href="https://sites.google.com/view/enricofatighenti/home">Enrico Fatighenti</a>
and
<a href="https://sites.google.com/site/fabiotanturri/home?authuser=0">Fabio Tanturri</a>
quickly produced a zero locus with the same properties,
using $\mathcal{Q}^\vee(1)$ on $\operatorname{Gr}(2,8)$.
<p>To find further evidence,
or disprove that they were the same variety,
we wanted to compute the degree of $\mathcal{O}_X(1)$
and the Hilbert series of $\mathcal{O}_X(1)$,
for which we needed the point class and Todd class.
<p>Turns out that those numbers agree,
thus giving further evidence to the conjecture that they are indeed the same variety!
Next up (please take up this challenge!) would then be:
<ul>
<li>trying to (dis)prove the conjecture,
<li>find similar descriptions for Kronecker quiver moduli
</ul>
<p>Et voilà,
now you know why I am so happy with this paper,
and you know what is in it, without having to read the less colloquially written introduction (or even worse, the paper itself).
Of course, you are still cordially invited to do so
(and let us know if you have any comments)
if you are interested in quiver moduli!
Thu, 06 Jul 2023 00:00:00 +0000
http://pbelmans.ncag.info/blog/2023/07/06/new-paper-chow-rings-quiver-moduli/
http://pbelmans.ncag.info/blog/2023/07/06/new-paper-chow-rings-quiver-moduli/algebraic geometrymathematicsFortnightly links (170)<p>This installment is so late that calling it monthly links wouldn't even salvage it... I've been writing papers instead of blogposts recently, so stay tuned for some updates on that front.
<ul>
<li><p><a href="https://arxiv.org/abs/2306.09908">Asher Auel, Avinash Kulkarni, Jack Petok, Jonah Weinbaum: A census of cubic fourfolds over $\mathbb{F}_2$</a>
does something really cool.
Back in 1914 <a href="https://mathscinet.ams.org/mathscinet/article?mr=1502501">Dickson classified (by hand) all smooth cubic surfaces over $\mathbb{F}_2$</a>.
There are 36 smooth cubic surfaces over $\mathbb{F}_2$.
Using automated methods, the classification of cubic 3-folds and cubic 4-folds over $\mathbb{F}_2$ is now also established.
There are 1 069 562 smooth cubic 4-folds!
And it is possible to do point and line counts, and all kinds of investigations.
A few weeks ago Jack also explained to me what is going to appear in the work-in-progress [3],
and I'm sure that will also be featured in the fortnightly links in due time.
<li><p><a href="https://unlocked.microsoft.com/ai-anthology/terence-tao/">Terence Tao: Embracing change and resetting expectations</a> is an essay on the role that AI might play in research mathematics soon. Highly recommended read!
<p>Maybe I can ask AI to write the fortnightly links for me soon? The benefit would be that they are on time, but all the fun would be lost, so rather not really!
<li><p><a href="https://www.erdosproblems.com/start">List of Erdős problems</a> is a nice interactive website
dedicated to all the problems (or at least, many)
that have been phrased by Erdős,
listing what is (not) known about them.
<li><p><a href="https://arxiv.org/abs/2305.18058">Anton Fonarev: Derived category of moduli of parabolic bundles on $\mathbb{P}^1$</a>
advertises a conjecture on the derived category of the moduli space of parabolic bundles on $\mathbb{P}^1$,
expanding on some discussion Anton and I had a few years ago.
I have some joint work-in-progress coming out, hopefully by the end of the summer,
generalising this conjecture (and the BGMN conjecture, or Tevelev theorem),
and I'm excited tell you all soon!
<li><p><a href="https://arxiv.org/abs/2305.17213v1">Alexander Kuznetsov, Evgeny Shinder: Derived categories of Fano threefolds and degenerations</a>
is a really cool preprint that explains how to fix one of the first conjectures in the theory of derived categories of Fano 3-folds,
explaining a relationship between Fano 3-folds of index 1 and 2 (in the Picard rank 1 case).
The introduction (and the rest) of the paper is a highly recommended read,
if you care about derived categories of Fano 3-folds.
</ul>
Mon, 03 Jul 2023 00:00:00 +0000
http://pbelmans.ncag.info/blog/2023/07/03/fortnightly-links-170/
http://pbelmans.ncag.info/blog/2023/07/03/fortnightly-links-170/fortnightly linksmathematicsFortnightly links (169)<ul>
<li><p><a href="https://www.math.uni-bielefeld.de/~hkrause/completion.pdf">Henning Krause: Completions of triangulated categories</a>
are lecture notes on (you'd never guess) completions of triangulated categories. It's amazing how useful completion has turned out to be in this subject, and these lecture notes are an excellent introduction.
<li><p><a href="https://arxiv.org/abs/2304.14048v1">Shinnosuke Okawa: Semiorthogonal indecomposability of minimal irregular surfaces</a>
proves that (minimal) surfaces whose $\mathrm{H}^1(S,\mathcal{O}_S)$ is nonzero (= irregular)
have an indecomposable derived category.
This rules out any interesting semiorthogonal decompositions for surfaces of general type,
except those for described explicitly in Corollary 1.10.
I don't know examples of such surfaces. Do you? Please tell me!
<li><p><a href="https://arxiv.org/abs/2305.06867">Warren Cattani: On the derived category of IGr(3, 9)</a>
constructs a full exceptional collection (in fact a minimal Lefschetz collection)
on the <a href="https://www.grassmannian.info/horospherical/X3(4,3)">horospherical variety IGr(3, 9)</a>.
Cool!
<li><p><a href="https://arxiv.org/abs/2304.12602">Geordie Williamson: Is deep learning a useful tool for the pure mathematician?</a>
gives both a gentle introduction to neural networks and deep learning,
and an overview of how it has been successful already in pure mathematics.
The question that I'm wondering about is:
when is the first big result in algebraic geometry coming from these methods?
Or have I missed something already?
</ul>
Wed, 17 May 2023 00:00:00 +0000
http://pbelmans.ncag.info/blog/2023/05/17/fortnightly-links-169/
http://pbelmans.ncag.info/blog/2023/05/17/fortnightly-links-169/fortnightly linksmathematics