Pieter Belmans
http://pbelmans.ncag.info/
Fri, 20 Nov 2020 08:51:03 +0000Fri, 20 Nov 2020 08:51:03 +0000Jekyll v3.9.0An update for Fanography.info: semisimplicity of quantum cohomology<p>Recall that Dubrovin's conjecture states the equivalence between
<ul>
<li>the existence of a full exceptional collection in $\mathbf{D}^{\mathrm{b}}(X)$
<li>the generic semisimplicity of (big) quantum cohomology $\mathrm{BQH}(X)$
</ul>
<p>for a smooth projective variety $X$. One can study Dubrovin's conjecture for special classes of varieties, and <a href="/blog/2020/11/06/qh-on-grassmannian-info/">I already added its status for $G/P$</a> to <a href="http://grassmannian.info">Grassmannian.info</a>.
<p>The conjecture is particularly interesting (and originally phrased for) Fano varieties, and for del Pezzo surfaces it is known to hold by Bayer's results. That brings us to Fano 3-folds, where I was wondering how much was known. Whilst figuring this out, I decided to add this information to <a href="http://fanography.info">Fanography.info</a>, and here we are.
<p>On <a href="https://fanography.pythonanywhere.com/dubrovin">Dubrovin's conjecture</a> I made an overview of the Fano 3-folds for which
<ul>
<li>expect (and in fact know) the existence of a full exceptional collection, for which a necessary condition (and in fact sufficient) condition is that the cohomology is of Hodge-Tate type, i.e. $\mathrm{h}^{1,2}=0$
<li>expect (and only <em>partially</em> know) the semisimplicity
</ul>
<p>It turns out that in all these (known) cases, the <em>small</em> quantum cohomology is already generically semisimple. At the time of writing there seem to be 17 Fano 3-folds for which semisimplicity is not yet known, and I'll try to make some progress on this. Please let me know if I've missed some literature.
<p><small>I should make things prettier at some point, and also add information on semiorthogonal decompositions. And figure out the original references for the quantum cohomology of $\rho=1$ Fano 3-folds. And add information on extremal contractions (a tiny piece of it is already available if you know where to look). Let me know if you have any other suggestions and requests!</small>
Fri, 20 Nov 2020 00:00:00 +0000
http://pbelmans.ncag.info/blog/2020/11/20/update-fanography-dubrovin/
http://pbelmans.ncag.info/blog/2020/11/20/update-fanography-dubrovin/algebraic geometryprogrammingmathematicsFortnightly links (117)<ul>
<li><p><a href="https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/blms.12429">Amnon Neeman: New progress on Grothendieck duality, explained to those familiar with category theory and with algebraic geometry</a> has a self-explanatory title. I approve of the title, and the content of the paper, as it beautifully describes aspects one of my first loves in mathematics.
<li><p>With the notion of a blogroll gone out of fashion for many, many years by now, I can amplify <a href="https://picturethismaths.wordpress.com">Picture This Maths</a> via these fortnightly links. They write very visually attractive blog posts!
<li><p><a href="https://arxiv.org/abs/2010.14371">Christian Böhning, Hans-Christian Graf von Bothmer, Roberto Pignatelli: A rigid, not infinitesimally rigid surface with K ample</a> has another self-explanatory title. What is unfortunate is that we can't add it to <a href="https://superficie.info">superficie.info</a>, because $\mathrm{c}_1^2=1260966$ and $\mathrm{c}_2=560658$ is slightly out of what we can display!
</ul>
Thu, 12 Nov 2020 00:00:00 +0000
http://pbelmans.ncag.info/blog/2020/11/12/fortnightly-links-117/
http://pbelmans.ncag.info/blog/2020/11/12/fortnightly-links-117/fortnightly linksmathematicsAn update for Grassmannian.info: semisimplicity of quantum cohomology<p>This blogpost does not serve as an introduction to Dubrovin's conjecture, let me just say that it states the equivalence between
<ul>
<li>the existence of a full exceptional collection in $\mathbf{D}^{\mathrm{b}}(G/P)$
<li>the generic semisimplicity of (big) quantum cohomology $\mathrm{BQH}(G/P)$
</ul>
<p>The first is a statement about the algebraic geometry of $G/P$, whilst the second concerns the symplectic geometry of $G/P$, and their equivalence is an amazing prediction motivated by mirror symmetry.
<p>One thing we can do is tabulate for which $G/P$ either is known to hold, and <a href="http://grassmannian.info">Grassmannian.info</a> now does that. Remark that, whilst the conjecture concerns the generic semisimplicity of big quantum cohomology, there is an easier version called small quantum cohomology, where computations are more tractable. And semisimplicity of the small version implies it for the big version, but it is not equivalent to it as there can be cases where the small quantum cohomology is <em>not</em> semisimple.
<p>Let me just end by saying that the data on the website is based on the table on page 33 of <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2821244">Chaput–Perrin: On the quantum cohomology of adjoint varieties</a> and for the generic semisimplicity of the coadjoint Grassmannians on unpublished work-in-progress of Chaput–Smirnov. I want to thank Maxim Smirnov for interesting discussions on the subject.
<p><small>The conjecture is phrased for all smooth projective varieties, but for the purposes of this blogpost we only care about $G/P$.</small>
Fri, 06 Nov 2020 00:00:00 +0000
http://pbelmans.ncag.info/blog/2020/11/06/qh-on-grassmannian-info/
http://pbelmans.ncag.info/blog/2020/11/06/qh-on-grassmannian-info/algebraic geometryprogrammingmathematicsRecordings and slides for the Felix Klein lectures by Markus Reineke<img src="/assets/fkl-reineke.jpg" width="300" style="float: right; margin: 10px">
<p>The <a href="https://www.hcm.uni-bonn.de/events/eventpages/felix-klein-lectures/fkl-2020-reineke/">Felix Klein Lectures</a> are over (they were great!), and all recordings and slides are available on the website.
<p>I want to thank Markus again, for giving such a great lecture series, which involved improvising through power and internet outages, and quarantine. Thanks also to Paul Wedrich for dealing with the on-site aspect of the lecture series (and hence improvising through power and internet outages).
Mon, 02 Nov 2020 00:00:00 +0000
http://pbelmans.ncag.info/blog/2020/11/02/fkl-recordings/
http://pbelmans.ncag.info/blog/2020/11/02/fkl-recordings/Two additions to 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>Just a quick service announcement: I have now added Hodge diamonds for
<ul>
<li>the Fano varieties of $i$-dimensional linear subspaces on the intersection of $Q_1\cap Q_2\subset\mathbb{P}^{2g}$
<li>the Grothendieck–Knudsen moduli space $\overline{M}_{0,n}$ of stable $n$-pointed rational curves
</ul>
<p>The first case, for $i=g-2$, is also the moduli space discussed in <a href="/blog/2020/10/26/canonical-strips-moduli-bundles/">the previous post</a>.
<p>The usual comment applies: feel free to suggest other varieties whose Hodge diamonds you would like to be implemented!
Wed, 28 Oct 2020 00:00:00 +0000
http://pbelmans.ncag.info/blog/2020/10/28/two-additions-to-hodge-diamond-cutter/
http://pbelmans.ncag.info/blog/2020/10/28/two-additions-to-hodge-diamond-cutter/algebraic geometrymathematicsPictures of the canonical strip hypothesis for moduli of vector bundles<h3>Golyshev's canonical strip hypothesis</h3>
<p>As a Christmas treat almost 2 years ago I produced some <a href="/blog/2018/12/26/toric-scatterplot/">colourful pictures of the roots of Hilbert polynomials of smooth toric Fano varieties</a>. These were meant to illustrate Golyshev's hypothesis, which predicts that these roots are to lie in the "canonical strip". Jointly with Sergey Galkin and Swarnava Mukhopadhyay we have found <a href="https://www.tandfonline.com/doi/full/10.1080/10586458.2019.1602571">counterexamples to this hypothesis</a> in higher dimension, using non-toric Fano varieties: namely moduli of vector bundles of rank 2 and odd determinant, on curves of genus $g$).
<h3>The two players</h3>
<p>There is an interesting parallel between
<ul>
<li>moduli of rank 2 bundles with odd determinant on a curve of genus $g\geq 2$
<li>moduli of <em>parabolic</em> rank 2 bundles on $\mathbb{P}^1$ with weight 1/2 at $2g+1$ points, for $g\geq 2$
</ul>
<p>The first is a Fano variety of Picard rank 1, index 2 and dimension $3g-3$, whilst the second is a Fano variety of Picard rank $2g+2$, index 1, and dimension $2g-2$. and more importantly, for both we have a <strong>Verlinde formula</strong> expressing dimensions of global sections of the anticanonical divisor. Hence we can determine their Hilbert polynomials, and compute the location of their roots.
<h3>The pictures</h3>
<p>Doing some totally unrelated reading on moduli of parabolic bundles a few days ago and coming across this Verlinde formula again, I wanted to (<small>I might suffer from some mild ADD?</small>)
<ul>
<li>check the possible failure of Golyshev's hypothesis for these moduli spaces of parabolic bundles
<li>make a colourful version of the plot we made in our paper and the analogous plot for the parabolic case
</ul>
<p>In the two plots below, once the roots go outside the interval $[-2,0]$ (resp. $[-1,0])$, the hypothesis fails. This happens for $g\geq 10$ (resp. $g\geq 8$). So this gives a second class of counterexamples! And in fact, whilst the counterexample we originally constructed had dimension 27, we now already see a counterexample in dimension 14.
<p>I think it would be interesting to understand the asymptotic behaviour which one observes in these pictures. Let me know if you have any suggestions!
<style type="text/css">
.axis path,
.axis line {
fill: none;
stroke: grey;
stroke-width: 1;
shape-rendering: crispEdges;
}
</style>
<script src="https://d3js.org/d3.v4.min.js"></script>
<div class="usual"></div>
<div class="parabolic"></div>
<script>
// setting the dimensions of the canvas
var margin = {top: 30, right: 30, bottom: 30, left: 30},
width = 600 - 30 - margin.left - margin.right,
height = width;
// setting the ranges
var x = d3.scaleLinear().range([0, width]);
var y = d3.scaleLinear().range([height, 0]);
// defining the axes
var xAxis = d3.axisBottom(x);
var yAxis = d3.axisLeft(y);
// adding the svg canvas
var parabolic = d3.select("div.parabolic")
.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 + ")");
// Get the data
d3.csv("/assets/golyshev-parabolic.csv", function(error, data) {
// scaling the range of the data
x.domain([-1.1, 0.1]);
y.domain([-2, 2]);
// color scale
color = d3.scaleLinear().domain([5, 15, 29]).range(["blue", "green", "red"]);
// the scatterplot
parabolic.selectAll("dot")
.data(data)
.enter().append("circle")
.attr("r", function(d) { return 2; })
.attr("style", function(d) { return "fill: " + color(d.n); })
.attr("cx", function(d) { return x(d.real); })
.attr("cy", function(d) { return y(d.imag); });
// axes
parabolic.append("g")
.attr("class", "x axis")
.attr("transform", "translate(0," + height + ")")
.call(xAxis);
parabolic.append("g")
.attr("class", "y axis")
.call(yAxis);
// legend
var legend = parabolic.selectAll(".legend")
.data([5,15,29])
.enter().append("g")
.attr("class", "legend")
.attr("transform", function(d, i) { return "translate(0," + i * 20 + ")"; });
legend.append("circle")
.attr("cx", width - 18)
.attr("transform", function(d, i) { return "translate(0, 9)"; })
.attr("r", 6)
.style("fill", color);
legend.append("text")
.attr("x", width - 30)
.attr("y", 9)
.attr("dy", ".35em")
.style("text-anchor", "end")
.text(function(d) { return d + " points"; })
});
</script>
<script>
// setting the dimensions of the canvas
var margin = {top: 30, right: 30, bottom: 30, left: 30},
width = 600 - 30 - margin.left - margin.right,
height = width;
// setting the ranges
var x = d3.scaleLinear().range([0, width]);
var y = d3.scaleLinear().range([height, 0]);
// defining the axes
var xAxis = d3.axisBottom(x);
var yAxis = d3.axisLeft(y);
// adding the svg canvas
var usual = d3.select("div.usual")
.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 + ")");
// Get the data
d3.csv("/assets/golyshev-positive-genus.csv", function(error, data) {
// scaling the range of the data
x.domain([-2.4, 0.4]);
y.domain([-2, 2]);
// color scale
color = d3.scaleLinear().domain([2, 17, 30]).range(["blue", "green", "red"]);
// the scatterplot
usual.selectAll("dot")
.data(data)
.enter().append("circle")
.attr("r", function(d) { return 2; })
.attr("style", function(d) { return "fill: " + color(d.g); })
.attr("cx", function(d) { return x(d.real); })
.attr("cy", function(d) { return y(d.imag); });
// axes
usual.append("g")
.attr("class", "x axis")
.attr("transform", "translate(0," + height + ")")
.call(xAxis);
usual.append("g")
.attr("class", "y axis")
.call(yAxis);
// legend
var legend = usual.selectAll(".legend")
.data([2,10,20,30])
.enter().append("g")
.attr("class", "legend")
.attr("transform", function(d, i) { return "translate(0," + i * 20 + ")"; });
legend.append("circle")
.attr("cx", width - 18)
.attr("transform", function(d, i) { return "translate(0, 9)"; })
.attr("r", 6)
.style("fill", color);
legend.append("text")
.attr("x", width - 30)
.attr("y", 9)
.attr("dy", ".35em")
.style("text-anchor", "end")
.text(function(d) { return "genus " + d; })
});
</script>
Mon, 26 Oct 2020 00:00:00 +0000
http://pbelmans.ncag.info/blog/2020/10/26/canonical-strips-moduli-bundles/
http://pbelmans.ncag.info/blog/2020/10/26/canonical-strips-moduli-bundles/algebraic geometryprogrammingmathematicsFortnightly links (116)<ul>
<li><p><a href="https://arxiv.org/abs/2010.08692">Mykola Matviichuk, Brent Pym, Travis Schedler: A local Torelli theorem for log symplectic manifolds</a> gives, aside from many important theoretical results, a classification of the deformation theory of $\mathbb{P}^4$. In particular, there are 40 different components (for $\mathbb{P}^2$ there is a single one, for $\mathbb{P}^3$ there are 6), and I'm looking forward to their more detailed analysis they have announced!
<li><p><a href="https://arxiv.org/abs/2010.01127v1">Daniel Halpern-Leistner: Derived $\Theta$-stratifications and the D-equivalence conjecture</a> is a long-awaited preprint of Daniel discussing an instance of the "birational Calabi-Yau implies derived equivalence" conjecture in <em>arbitrary</em> dimension (it is known in dimension 3 by Bridgeland), for varieties which are birational to moduli spaces of sheaves on K3 surfaces. Bridgeland comes into this picture again, now by virtue of Bridgeland stability conditions, as the proof goes via wall-crossing. In Bonn we've run a seminar on this topic 2 years ago, and it is a really interesting machinery behind the proof.
<li><p><a href="https://arxiv.org/abs/2010.08976">Tanya Kaushal Srivastava: Pathologies of Hilbert scheme of points of supersingular Enriques surface</a> shows that Hilbert schemes of points on supersingular Enriques surfaces (in characteristic 2 the $\mathbb{Z}/2\mathbb{Z}$ one has as torsion in the Picard group can be come either $\mu_2$, the <em>singular</em> case or $\alpha_2$, the <em>supersingular</em> case) are examples of varieties showing that the pretty description of the Hodge numbers fails, and more importantly that the Beauville–Bogomolov classification of varieties with trivial canonical bundle and no étale covers is richer than in characteristic 0. Cool!
</ul>
Sun, 25 Oct 2020 00:00:00 +0000
http://pbelmans.ncag.info/blog/2020/10/25/fortnightly-links-116/
http://pbelmans.ncag.info/blog/2020/10/25/fortnightly-links-116/fortnightly linksmathematicsRegistration for Felix Klein lectures by Markus Reineke<img src="/assets/fkl-reineke.jpg" width="300" style="float: right; margin: 10px">
<p>The <a href="https://www.hcm.uni-bonn.de/events/eventpages/felix-klein-lectures/fkl-2020-reineke/">registration for the Felix Klein lectures by Markus Reineke</a> is now open. Markus is a great lecturer, and he will talk about very exciting topics, so I strongly suggest anyone who might be interested in representation theory or algebraic geometry to register!
<p>For good measure, here's the abstract again:
<blockquote>
<p>Quiver moduli spaces are algebraic varieties encoding the continuous parameters of linear algebra type classification problems. In recent years their topological and geometric properties have been explored, and applications to, among others, Donaldson-Thomas and Gromov-Witten theory have emerged.
<p>The first aim of the lectures is to motivate the study of quiver moduli spaces from the point of view of representation theory, to review their construction via Geometric Invariant Theory, and to discuss several classes of examples. We will proceed by reviewing results on the topology and geometry of these moduli spaces, in particular their cohomology.
<p>Finally, we will discuss applications of quiver moduli spaces to Gromov-Witten and Donaldson-Thomas theory via wall-crossing.
</blockquote>
Fri, 16 Oct 2020 00:00:00 +0000
http://pbelmans.ncag.info/blog/2020/10/16/fkl-registration/
http://pbelmans.ncag.info/blog/2020/10/16/fkl-registration/Fortnightly links (115)<ul>
<li><p><a href="https://arxiv.org/abs/2009.13382v1">Lorenzo De Biase, Enrico Fatighenti, Fabio Tanturri: Fano 3-folds from homogeneous vector bundles over Grassmannians</a> gives alternative descriptions of the 105 families of Fano 3-folds, all in terms of zero loci of homogeneous vector bundles on products of Grassmannians. The nice thing about this approach is that it allows for extensions to the case of Fano 4-folds, which the authors are working on. They told me they are finding hundreds of unique combinations of invariants, which should then be cross-checked with the existing constructions, and hopefully yield many new Fano 4-folds!
<p>This is the third way of describing Fano 3-folds, after the birational approach due to Mori–Mukai and the approach used by Coates–Corti–Galkin–Kasprzyk (using a combination of toric geometry and vector bundles, to which known quantum cohomological methods could be applied).
<p>Only the Mori–Mukai approach is featured on <a href="http://fanography.info">Fanography</a>, meaning that I should add these two alternative descriptions soon!
<li><p><a href="https://arxiv.org/abs/2009.12630v1">Will Donovan: Stringy Kähler moduli for the Pfaffian-Grassmannian correspondence</a> explains the suggestion from physics that the monodromy of the Kähler moduli space gives autoequivalences of Calabi-Yau varieties. For the case of the Pfaffian-Grassmannian equivalence (a derived equivalence between non-birational Calabi-Yau threefolds, meaning that they are "double mirrors") this leads to beautiful visualisations of the moduli space and the autoequivalences it induces. Cool stuff!
<li><p><a href="https://www.quantamagazine.org/building-the-mathematical-library-of-the-future-20201001/">Building the Mathematical Library of the Future</a> is a Quanta article about how Lean is being used to set up a library of results and proofs, on which more complicated proofs can then be built. There exists <a href="https://leanprover-community.github.io/mathlib_docs/">an overview of what mathlib</a> knows. Especially the <tt>algebraic_geometry</tt> section is interesting, and we hope that at some point there will be some interaction between mathlib and the Stacks project.
</ul>
<p>Maybe I should start mentioning in fortnightly links when Kerodon was updated. Today, <a href="https://kerodon.net/tag/01W4">a section on $(\infty,2)$-categories was added</a>.
Sun, 11 Oct 2020 00:00:00 +0000
http://pbelmans.ncag.info/blog/2020/10/11/fortnightly-links-115/
http://pbelmans.ncag.info/blog/2020/10/11/fortnightly-links-115/fortnightly linksmathematicsInterpreting skew conics<p>In <a href="/blog/2020/08/11/fortnightly-links-111/">the 111th installment of fortnightly links</a> I mentioned <a href="https://arxiv.org/abs/2008.02255v1">Derived categories of skew quadric hypersurfaces</a> by Kenta Ueyama. In this paper he discusses a class of noncommutative objects whose definition mimicks that of the usual quadrics, but the ambient space is now a certain skew noncommutative projective space. For this reason he calls them <em>skew quadrics</em>.
<p>And already in (projective) dimension 1 these skew quadrics are not necessarily deformations of usual quadrics (even though their ambient noncommutative projective space is), because their derived categories don't have the same number of objects, and this is a deformation-invariant. Ueyama shows there are precisely two types which appear in this way.
<p>I told you that I find this an intriguing picture that I would love to understand better. Soon after posting the fortnightly links I came up with an interpretation in the case of dimension 1 (so <em>skew conics</em> if you wish) which allows me to explain why there are these two cases. I'm sharing this here on my blog now in case someone finds this $\epsilon>0$ interesting, and I hope it could spark some further discussion. It's also been a while since I've done a longer mathematical blogpost (although I realise that this one will not appeal to a general audience :)).
<h3>Skew conics</h3>
<p>Let us recall the setup from Ueyama's preprint. A <em>skew polynomial algebra</em> is the graded algebra $A_q=\mathbb{C}_q[x_0,\ldots,x_n]$ where $x_ix_j=q_{i,j}x_jx_i$ for some symmetric matrix $q=(q_{i,j})$ such that $q_{i,i}=1$. To such an algebra (and in fact any quadratic Artin--Schelter regular algebra, of which this is just a specific instance) we associate the category $\operatorname{qgr}A_q$, which is a noncommutative projective space.
<p>If $q_{i,j}\in\{\pm1\}$ for all $i$ and $j$ then the Fermat quadric $x_0^2+\ldots+x_n^2$ is a central element, which we'll denote by $f$. Then the category $\operatorname{qgr}A_q/f$ is a <em>skew quadric hypersurface</em>. It is studied using algebraic methods in Ueyama's preprint, and in particular the structure of its derived category is described. For $n=1,2,3$ a classification is also given. The case $n=1$ being not very interesting, let us consider $n=2$. Hence we have a skew quadric hypersurface inside a noncommutative projective plane, or <em>skew conic</em>. It is shown in Example 3.23 that its derived category is one of the following:
<ul>
<li>if $q_{1,2}q_{1,3}q_{2,3}=1$, then $\mathbf{D}^{\mathrm{b}}(\operatorname{qgr}A_q/f)$ is the derived category of the Kronecker quiver (denoted as extended $\mathrm{A}_1$ in op. cit.), so $\mathbf{D}^{\mathrm{b}}(\operatorname{qgr}A_q/f)\cong\mathbf{D}^{\mathrm{b}}(\mathbb{P}^1)$;
<li>if $q_{1,2}q_{1,3}q_{2,3}=-1$, then $\mathbf{D}^{\mathrm{b}}(\operatorname{qgr}A_q/f)$ is the derived category of the extended $\mathrm{D}_4$ quiver.
</ul>
<h3>Interpretation</h3>
<p>What is a geometric interpretation for this second case? For starters, this dichotomy should be related to the geometry of the point scheme of the ambient noncommutative projective space, see <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3527537">Belmans–De Laet–Le Bruyn: The point variety of quantum polynomial rings</a> for this story (including higher dimensions, which we'll get back to later).
<p>To make this precise for $n=3$, notice that the assumption that $q_{i,j}\in\{\pm1\}$ causes the skew polynomial algebra to be a <em>Clifford algebra</em>, and that such noncommutative projective planes can be interpreted as quaternion orders, as discussed in <a href="https://arxiv.org/abs/1811.08810">Belmans–Presotto–Van den Bergh: The Hirzebruch isomorphism for exotic noncommutative surfaces</a>. Indeed, for $q_{1,2}q_{1,3}q_{2,3}=-1$ the skew polynomial algebra turns out to be finite over its center, whose central Proj is $\mathbb{P}^2$, and we obtain a sheaf of orders. This is a sheaf of (noncommutative) algebras $\mathcal{A}$ which is generically an Azumaya algebra, except on the ramification divisor, which is a triangle of lines in this case.
<p>This brings us to the <strong>main idea</strong>:
<blockquote>
a skew conic is the restriction of a quaternion order to a line!
</blockquote>
<p>Indeed, one can reinterpret the Fermat quadric $f$ as a section of $\mathcal{O}_{\mathbb{P}^2}(1)$ on the central Proj. If we restrict the quaternion order to this line, it will intersect the ramification divisor in 3 distinct points. We obtain a quaternion order (which is no longer maximal, only hereditary) on $\mathbb{P}^1$, whose ramification are 3 points.
<p>But this is the same as a weighted projective line using the <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2018958">Chan–Ingalls dictionary</a>! And these derived categories are well-understood. The weighted projective line we obtain in this way is "of type $(2,2,2)$" (as it is a quaternion order, and its ramification divisor consists of 3 points), and its derived category is equivalent to that of the extended $\mathrm{D}_4$ quiver.
<p>Written like this I think this is a satisfactory geometric interpretation of skew conics.
<h3>Higher dimensions?</h3>
<p>But what about higher dimensions? In section 4 of his preprint Ueyama obtains 3 possibilities for the structure of the derived category of a skew quadric surface. One is the familiar $\mathbf{D}^{\mathrm{b}}(\mathbb{P}^1\times\mathbb{P}^1)$ (denoted QS1), the other two (see Proposition 4.2 and ensuing discussion) have are different. For QS2 we recognise a noncommutative quadric surface, whilst QS3 has a quiver with 10 vertices.
<p>This blogpost has been rambling for long enough; let me just point out that the classification of point schemes for skew polynomial algebras for $n=4$ gives 4 possibilities, but one of them can not be realised using $q_{i,j}\in\{\pm1\}$. Also, the central Proj in this case is more complicated, one does not just get $\mathbb{P}^3$, but a more complicated singular rational 3-fold. On this 3-fold we should get a natural hyperplane section, such that the restriction of the quaternion order is the skew quadric surface. I don't know how to make the story more precise, I don't even know what the hyperplane section looks like. Is it isomorphic to $\mathbb{P}^1\times\mathbb{P}^1$? How does it hit the ramification? What is the ramification? Many interesting questions!
<p><small>Is this something that I should write up with more details and actual proofs? Coming up with this interpretation was certainly aided by the papers I wrote during my PhD, so many details omitted here are based on things I thought about at that time, which are not necessarily "standard" material. I also hope I didn't mess anything up.</small>
<p><small><strong>Acknowledgements</strong> I want to thank Theo Raedschelders for listening to me rambling about this in emails. If you think this blogpost was rambling, be thoughtful of what Theo went through.
Tue, 29 Sep 2020 00:00:00 +0000
http://pbelmans.ncag.info/blog/2020/09/29/interpreting-skew-quadrics/
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