Video summary
The seminar introduces fundamental concepts in algebraic geometry, focusing on moduli spaces that classify geometric objects up to equivalence and sheaves which generalize vector bundles to accommodate singularities. A central theme is the Donaldson-Uhlenbeck-Yau theorem, establishing a deep link between algebraic stability conditions for holomorphic vector bundles or locally free sheaves and the existence of Hermitian-Einstein metrics solutions to partial differential equations. The discussion transitions from well-behaved moduli spaces on curves to more complex structures on surfaces and threefolds like $\mathbb{P}^3$, where higher-dimensional spaces often exhibit multiple irreducible components, varying dimensions, and singularities described by Vakil's Murphy Law. Recent research highlights the structure of rank-two sheaves on $\mathbb{P}^3$ with maximal third Chern class ($C_3$), revealing that stable reflexive sheaves exist only when $C_3 \leq C_2^2$, while specific transformations involving global sections allow for the construction of new families of irreducible components in torsion-free sheaves, effectively lowering $C_3$ and generating novel geometric structures.
Stability conditions are rigorously defined through Hilbert polynomials, distinguishing between $\mu$-stability, Gieseker-stability, and semi-stability, with a clear hierarchy where $\mu$-stable implies stable, which in turn implies semi-stable. In the specific context of rank two sheaves on $\mathbb{P}^3$ with first Chern class $-1$, all stability notions coincide for $c_2 \geq 3$, ensuring that every semi-stable sheaf is fully stable without proper subobjects. The analysis utilizes the spectrum of sheaves to determine cohomology groups via Euler characteristics, proving that torsion-free sheaves in this range admit non-zero global sections. This property renders all pairs delta-stable and very stable under a stability notion defined by inequalities involving reduced Hilbert polynomials perturbed by $\Delta$, eliminating intermediate walls between saturated and non-saturated states found in other contexts.
The geometric implications of these findings are illustrated through the mapping of moduli spaces to specific families of curves, such as twisted cubics associated with $c_2=3$ or planar cubics plus a point for higher values where reflexive sheaves vanish. For instance, when $c_2 \geq 4$, only the component corresponding to planar curves and points remains, while at $c_2=3$, the Hilbert scheme comprises two distinct irreducible components linked to twisted cubics and planar configurations respectively. Although Schmidt has demonstrated smoothness and irreducibility for higher degrees using alternative methods, extending these results to products like $\mathbb{P}^1 \times \mathbb{P}^2$ or computing intersection theory on Hilbert schemes remains a significant challenge due to the inherent complexity of understanding the underlying curve Hilbert schemes. Ultimately, the seminar concludes by emphasizing how correspondence between bundles with sections and curves allows for a precise classification of moduli space components based on their geometric realizations as families of plane curves plus points.
Read the full video transcript
Okay, you're clear to start.
So, thanks everyone for showing up. I
was asked to do a
basic notions talk. So, I'll try to
explain a few things
behind the the work that I do in the
double research that I do in algebraic
geometry.
So, I'm assuming that some of you,
perhaps most of you are not familiar
with these words, moduli spaces,
sheaves. So, I'll try to first explain
what a moduli space What is the idea
behind a
moduli space and the idea behind this
notion of sheaves. And end up with
some of the results that we have proved
on this topic in the past couple of
years.
Uh try to click on the screen once.
Maybe it has to be activated.
With the With the mouse, just points on
the
Exactly, give it one click and now try
using the pointer. Yeah. Okay. Okay.
So, in in geometry
or in mathematics in general, right? So,
mathematicians like to classify stuff.
The
The dream of any any mathematician is to
say, "Okay, so any object of that type
is in this list."
Sometimes you can make a finite list or
some
Even if the list is not finite, you can
do a neat list like the classification
of
of finite abelian groups.
You know, for every
Let's say this the the order of the the
you can tell how many groups there are,
and this is the final list.
Um also in representation theory, you
can classify representations of um
of [clears throat] certain groups.
Um
And the in in geometry also have this
type of classification problem,
uh classification [clears throat]
of geometric objects, except that uh
more often than not, the objects we are
trying to classify are not finite.
They come in infinite families.
>> [clears throat]
>> And uh
we want to study collection of objects
like curves, uh manifolds, or vector
bundles in the in uh I'll explain what
these are in a moment,
which is what I do. And the the answer
that algebraic geometry particular gives
to this classification problem is called
a moduli space.
Um so a moduli space is some sort of
geometric space. This is in the broadest
sense of the word uh or a space,
uh in which each point corresponds to
one class one isomorphism class of the
objects you want to uh to to classify.
Right?
So we start with a collection of some
objects,
um
a big bag full of stuff inside,
um and then you need also to define when
two objects are the same, when they are
equivalent in some sense.
And uh as I said, the moduli space is
just this uh set of uh equivalence
classes of these objects you're trying
to to classify.
In such a way that each point in this
space corresponds to a unique
a unique [clears throat]
uh equivalence class, right? So you can
find a representative for each point,
you can find an object, and then uh
that the equivalence class of that
object will uniquely be assigned with
that point.
Um and in addition, we would like this
space, this moduli space, to be equipped
with some sort of geometric structure.
So that we can do geometry on the space
that classifies geometric objects.
So in this sense,
this is the geometry of geometry, right?
So we want to [snorts] do geometry in a
space that classifies geometric objects.
Let me give you a very simple example.
Uh
say you want to classify all
triangles in a plane, right? In the
regular Euclidean plane, you want to
classify all triangles up to congruence.
Right? So up to congruence. So one way
to define a congruence class of a
triangle is give uh
two
sides,
right?
And an angle between them.
So you can think of this space of all
congruence classes of triangles to be
uh
well, I have to give three parameters,
right? So one positive number for the
side of one
of one of the sides,
another positive number, plus an angle,
right? So plus something in the interval
between 0 and 180°.
So the product of these three
uh things will be the set of all
congruence classes of triangles.
Okay?
Because for any
any triangle, we have this
uh uh
side-angle-side
case of congruence of triangles and I
then I can find the unique address of
any given triangle in this address book.
So, you can think of a the moduli space
as an address book and the and the
coordinates here as the address of the
particular object you're trying to to
classify.
And now in this space,
right, which is some sort of strip,
sort of a I don't know.
Sort of a three-dimensional
space.
Strip like like this
that comes off infinitely from the
from the board,
you can start to do paths, right? So,
you can walk around also in this in this
space and
walking around in this space, you are
actually doing a family of triangles.
So,
you are deforming the triangle when one
of the legs of the triangle increases,
perhaps one of the the angles of the
triangle increases and then decreases.
So, you you're building you're you're
moving your triangle around. And this
movement of the triangle can be
translated as a path in this in this
space.
R plus times R plus times uh
Let me repeat. The second arc length is
the second length to be bigger or
smaller than this one.
No, because I'm I can give any sides and
the angle between them.
So, this always gives me a triangle.
Yeah, you know, it's just switch the
two. Yeah, okay, I'm lying a little bit
because I have to attach labels to to
these. Otherwise, you have these
automorphisms and I didn't want to to to
say that, but since you
Yeah,
but yes, you can make one smaller than
the other, but when they are equal, uh
you have this ambiguity ambiguity.
Otherwise, then you have to say side
number one and side number two and uh
kill the automorphisms.
>> [clears throat]
>> Good.
>> [snorts]
>> Now, what is a a vector bundle?
A vector bundle now is a family of
vector spaces parametrized by some uh
by some manifold, by some geometric
object. For me, the manifolds will be
projective algebraic varieties.
So, what is that? That is the common
zeros of a homo- of homogeneous
polynomials.
So, I take uh a bunch of uh homogeneous
polynomials
in [snorts]
n plus one variables.
These uh kappa here
is some field that is the uh
that gives me the coefficients of these
polynomials.
Okay, so I have a bunch of homogeneous
polynomials with coefficients in kappa,
right? And uh n plus one variables, and
the common zeros of these polynomials
gives me a projective variety.
And this is a
uh
And And that's what we are going to call
space for the purposes of this talk.
And uh
a vector bundle over an R is something
that uh is well, a family of vector
spaces. So, for each point, I attach a
vector space uh to that point
in such a way that it looks locally like
a product, right? So, for every open set
or there is uh a covering, right?
Um
I can find a a covering of X by open
sets that I'm going to denote
so that uh the uh the pre-image of this
open set looks like a product.
That's what they say when they say that
this vibration here, this is a
projection map, is locally trivial.
Um
but once you you know that locally it
looks like just a product, you have to
know how to glue them up in the varying
the various intersections of the
uh
of the open sets.
And these gluing are are done by these
so-called
transition functions, which are just
polynomial functions from the
intersections
of the various sets in the covering to
this group. That tells you how to change
the base, right? So, this is a GL
GL here is supposed to be how how you
move, right? So, how do you glue or
compare two different uh vector vector
spaces
of the same dimension n
or R. They are used R.
>> [snorts]
>> So, informally, vector bundle is a
family of vector spaces parameterized by
X in such a way that locally it looks
like a product.
Um and these admit a topological
classification in terms of some
invariants called the Chern classes,
right? I'm not going to explain what a
Chern class is, but it's some kind of
topological invariants that classifies
these
these invariants up to
uh
up to topology.
So, why care, right? So, vector bundles
are very important for various things,
in particular in remaining geometry and
the non-linear analysis on on manifolds.
Right? So, they are the right
mathematical language to describe
general relativity and many things in
physics,
like gauge theory and string theory. So,
it turns out that it's the right
mathematical object in
on which over which you can do physics
on manifolds. And physics here is both
high energy physics and also general
relativity.
Sheaves, right? So, a sheaf will be a
A sheaf is something that you can think
of a sheaf as a vector bundles with
singularities. A vector bundles that is
singular at some points.
Now, how do we go from a vector bundle
to a sheaf? So, for each
open covering
I can look at the set of all local
sections.
And by that I mean maps from the open
set
to the
to the product. So, maps from each open
set in the covering to the product of
this open set with the the vector space
kappa R copies.
And that in such a way that the image of
some point in X will be that point in X
plus a vector.
And this vector changes with with X.
Right? So, you can think of if you have
a family of vector spaces, this is a
sections, these local sections is a
family of vectors in those vector
spaces.
Right? So, so as you move around in your
space, the vector is also moving around
in the various vector spaces.
Well, but this
oops, yeah. But this vector
is
actually well, this F is a function from
the
the open set to the vector space
capital R
and
and it actually can be regarded as R
functions from the open set to the
field.
So this field can be the reals or the
complex numbers
for instance.
And these are polynomial maps. So that's
another way to say this fancy word
regular maps, right? So you can think of
a polynomial polynomials from the from
your open set which is
which you can regard as a some CN or an
RN if you prefer to
to R. So it's a bunch of polynomials.
Okay.
Um
which means that each set of all local
sections on an open set which is this
curly E
of UI,
the set of all local local sections on
an open set UI
is a free module over the set over the
ring of regular functions, the ring of
polynomial functions.
>> [clears throat]
>> So together all these uh
modules of local sections, they form
what is technically called a sheaf, a
locally free sheaf of OX modules.
And more generally you can consider
sheaves, right? So collections of
of
uh,
of modules over each open set, so
collection of
a collection of, uh, all UI modules.
Uh, but instead of these, each OUI
modules to, uh, to be free as as it
occurs in the case of vector bundles,
they can be more general modules like
reflexive or torsion-free modules.
Right, a reflexive module is a module,
uh, over a ring that is isomorphic to
its double dual.
And a torsion-free module is a module
that injects into into its double dual.
There's always a map from any module to
its double dual. When this map is
injective, we call that module a
torsion-free module. And when this,
uh, natural map into the double dual is
an isomorphism, we call it reflexive.
So, if you if you allow for these more
general modules instead of just, uh,
free modules over, uh, for each open
set, you get what is called, uh, you
know, a sheaf, a reflexive or
torsion-free sheaf.
Uh,
and these are the objects that we'd like
to classify. And coherent, I'm not going
to explain the meaning of this word,
coherent,
uh, sheaves of OX modules over
projective varieties. Right, so
something that to each open set in your
projective variety associates a module
over the ring of regular functions,
um,
on that open set.
Good. Uh, and, uh, moduli spaces of
sheaves, um,
which is, you know, now I want to
classify all such sheaves up to some,
uh, equivalence, some isomorphism or
some other equivalence, actually.
But we know that they, uh, that there
are some, uh,
as I mentioned, there are some
discrete, uh, topological invariants,
which are these Chern classes. So, the
first thing we do is fix the Chern
classes.
This is fixing, I don't know, like the
zip code of the Once you fix the zip
code, there's still more bits to the
address. So, the Chern classes are the
first invariants that you fix.
Uh,
once you know the the fixed code the the
the the zip code, there you still have
to go around in the different streets to
look for, uh, the the the correct house,
the correct address,
uh, you want to go.
Uh,
and it's been proved in the,
started with, uh, Mumford, David
Mumford, in the late, uh, '60s and then
continued in the '70s and early '80s,
that these moduli spaces actually have
the structure of a projective scheme.
Right? So, it's, uh, they are also, they
can also be, uh, regarded as a zeros of
a bunch of, uh, homogeneous polynomials.
So, this is the small, uh, miracle. It
It didn't have to be that way. It's a
small miracle of nature that you're
classifying these objects and they are
again the type of objects that you
started with in some sense. They are
again, uh, projective varieties or at
least projective schemes.
Um,
one very, uh,
very well-known result in this area is
the so-called Donaldson-Uhlenbeck-Yau,
uh, theorem that provides a bridge
uh, between differential geometry and,
uh, mathematical physics with, uh,
algebraic geometry in the middle.
So, a holomorphic vector bundle, so,
that's the statement of the
Donaldson-Uhlenbeck-Yau theorem, that
any holomorphic vector bundle or locally
free, uh, sheaf on a smooth compact
killer manifold is uh is slope stable.
This uh
it's some some property some algebraic
property of a locally free chief if and
only if it admits a compatible hermitian
Einstein metric. So, it's uh telling you
that uh some bit of
despite the technical word here is that
this statement telling you that some bit
of
uh
algebraic geometry is some uh is the uh
necessary condition for the existence of
solutions of a partial differential
equation on a manifold, right? Something
that is very scary and uh horrible,
right? So, solving partial differential
equations on curved spaces is very
scary, but this algebraic uh condition
stability tells you or uh uh
uh gives you a necessary condition for
the existence of solutions
for these uh
scary PDEs.
So, this is one example why
uh sheaves and and moduli spaces of
sheaves uh became uh
fashionable and important
um
from the more or less the
the '80s
uh
so on so forth the past 40 to 50 years.
Uh and this is also the bridge with uh
string theory and a bunch of
mathematical physics that is done
nowadays, right? Because uh
it turns out that the moduli space of uh
uh
stable sheaves
is uh the same as the
as the moduli space of solutions of a
given PDE, right? So, we can classify
solutions of a PDE using algebraic
geometry.
However, uh
uh these moduli spaces can be very nasty
um
very nasty spaces, right? They are even
in
uh
It's not because they are there they are
particularly nice. And uh even proving
that they are non-empty, proving the
existence of uh
of a stable uh bundles uh with given uh
turn classes and uh given you know that
there is a a blue house in a given uh
for a given zip code, right? Is not a
completely trivial uh matter, right? So,
you have to walk in all the streets to
check if one of these houses are is
blue. So.
>> [clears throat]
>> Good. So, let me uh
now uh start talking about some of the
the results that are known, you know,
the in the in the case of of curves,
which are uh projective varieties of
dimension one, this is the simplest
case, and and that's uh the situation
where you can tell the most about these
spaces.
So, I'm going to use this notation here
uh to mean the moduli space, right? So,
the set of all uh
semi-stable vector bundles
uh of rank R and degree D, right? So,
this D here is playing the role of the
turn class. Now, there's only one turn
class that is important, right? It uh
this turn class is called the degree,
and this is the invariant we fix for
curves.
Never [clears throat] mind what
semi-stable is. It's just some property
of a vector bundles. Uh we actually need
need this property in order to classify
because classifying all bundles
um doesn't give you a very good
geometric structure, but classic but if
you throw out some uh bad bundles, some
uh
uh bundles that are called unstable, and
you keep just the semi-stable ones, then
uh this uh moduli space this classifying
problem, this classification problem is
solved by a uh a decent uh variety.
Uh
so, this is always a a
uh a projective variety, right? And this
is irreducible projective variety of a
given dimension that that depends on the
um
rank and the genus of the curve. It
should depend on the degree also.
No? Oh, yeah, this is right. Yeah.
Sorry.
Yeah. So, that depends only on the rank
and the genus of the curve, which is
this G.
Uh
and this is for this situation where
when the when uh
rank and degree are coprime.
So, when rank and degree are coprime,
these uh projective varieties in
addition uh smooth.
So, this is the uh best possible
situation in which your objects are
classified by a by some neat object as
well. So, some smooth uh space. Some
smooth projective variety.
Um
And then uh there are some some uh there
is a very influential paper by Atiyah
and Bott from the um mid-80s that tells
you how to compute the cohomology of
these spaces.
So, you can tell something about the
topology of these spaces.
Um
The Narasimhan-Seshadri theorem provides
you a bridge with uh representation
theory
telling that uh stable bundles of degree
zero
uh correspond to irreducible
representations of the uh fundamental
group of the curve.
So, this is an another uh
unexpected link between
uh a little bit of representation theory
representing these uh these these
particular groups um
fundamental groups of uh Riemann
surfaces into uh
into unitary uh groups, right? Into UN.
They are uh
they are related to these bundles, and
they are classified by the same uh class
of objects. So, again, it's not only the
classification of geometric objects, but
also the classification of algebraic uh
objects. You're you're also classifying
representations of a certain group
uh
in uh
the space of unitary matrices, the group
of unitary matrices, for instance.
And uh
And Donaldson also gives a this is uh
you know, the the first version of the
Donaldson-Uhlenbeck-Yau theorem, which
it gives uh
uh the existence of a unitary connection
uh with constant curvature on the the
stable bundles. So, these objects are
actually in the middle of uh a bunch of
other things, representation theory,
differential geometry, algebraic
geometry. They are uh really uh
quite important uh
in in in
in maths in general, because there are
many things that you can relate to these
moduli spaces of sheaves on a curve.
Even if you're not interested in
algebraic geometry, this might be a a
nice uh bit of mathematics to learn.
Okay.
So, the moral of the story is that on
curves, moduli spaces are well-behaved,
and it's possible to study their
topology and geometry through algebraic
means, right? So, we can really
understand the geometry of the space
that classifies these bundles
uh on curves.
The situation is less rosy for surfaces,
but still um
for some in class of surfaces, there is
a lot you can say about these are moduli
spaces.
Now, we need
not only the rank
and and but also two turn classes that
we have I'm going to call C1 and C2.
Um
these are again the the zip code the the
the topological invariants of the
sheaves.
Um
And you can say something about the
existence.
So, when
this is always non-empty when C2 is
sufficiently large and how large depends
on the surface.
We have some
some some estimates given by these are
so-called Bogomolov inequality.
Um
When S is a special kind of surface
called a funnel surface.
So, that's the
So, if you tell me if you give me more
information about the surface, then I
can tell you more about the moduli space
of sheaves on the surface.
Uh
so, on a funnel funnel surface, these
guys are always smooth.
So, these are again an instance in which
um
there's a decent classification for
these um
on a surface.
Uh
and if you specify the surface a bit
more like a K3 surface, which is
specially kind of nice kind of surface,
you can tell more about geometric
structures on this moduli space because
this is another trend
trend also in in algebraic and
differential geometry. Okay, so you give
me you tell me what the space are.
So, now I can you tell me what the
moduli space is. Uh
now, let me do geometry on this moduli
space. And doing geometry is finding
different different types of geometric
structures um
uh on uh, this moduli space.
So, in particular, uh, Mukai in this,
uh, very important, uh, example tells
you that these moduli spaces are
hyperkähler manifolds.
Which is maybe, uh, the type of, uh,
geometric structure that is the most
rigid and and then has the most, uh, uh,
amount of information needed in order to
define what is a hyperkähler,
um,
manifold.
So, uh,
these the moral here that I that that
I'm trying to, uh, to to say is that,
uh, these moduli spaces they answer a
classification problem, but they are
also very interesting, uh, varieties on
their own, right? Because they also
have, uh, very rich, uh, geometric
structures and, uh, and and and
properties, topological properties and
geometric properties that are somehow
inherited [clears throat]
and can be, uh,
and can be read from the, uh, original
manifold that you started with.
>> [snorts]
>> So, they're also good ways to produce
new, uh, interesting examples of
varieties and, um,
and geometric structures on manifolds.
Um, one very interesting bit of theory
that was done in the in the the early
'90s is the the the so-called, uh,
topological, uh, Donaldson invariants,
right? So, they,
um,
>> [clears throat]
>> So, the the the the the the story's more
or less like this. The Donaldson
invariants are sort of, uh, second order
topological invariants of your original,
uh, surface. Start with a surface and
then you make this other space, a moduli
space, uh,
you make this other, uh, projective
variety, this, uh, moduli space of uh,
of bundles on a surface.
Uh
now you look at the topological
invariants of this new space.
And it turns out that this the
topological invariants of this new space
are also topological invariants of the
original space.
So going to to the moduli space is also
a way to produce new topological
invariants of your original manifold.
And Donaldson was able to
to prove very ground many groundbreaking
um
results in differential topology using
this this technique. All right, so
starting on a manifold, looking at a
moduli space of stuff on the original
manifold, do topology on this new space,
and discover something about the
topology of the original manifold.
Discover some highly non-trivial
topological invariants of the original
manifold that you started with.
So in in this sense, the moduli spaces
became a tool to understand the
differential topology, to understand the
algebraic geometry of the original
manifold that you that you were
interested in.
Uh
now, for varieties of higher dimension,
dimension greater than two, all of the
questions above are uh open and often
challenging.
All right, so and this is and this is
the threefold. So I explained what a
moduli space is. I said a little bit
about sheaves. Now let let me get to the
threefolds. So a threefold is a
projective variety of dimension three.
Okay.
Um
And here I'm I'm going to restrict
myself to my personal favorite to
threefold which is the P3.
Uh which is the the simplest possible
the projective spaces are the simplest
possible projective varieties, right?
Um
>> [clears throat]
>> and it's the
it's the situation where uh
as far as I know, you can say the most
about uh these these moduli spaces.
The fact is that these moduli spaces are
neither irreducible nor smooth in
general. Irreducible here means that
whether or not the moduli space
uh has a has one or various pieces.
Something that looks like
uh
like this, the union of two lines, this
is not irreducible, right? Because it
has two two bits.
And these
These are called irreducible components.
And these irreducible components
they also have different dimensions.
Like a plane with a straight line stuck
on it.
Uh
>> [clears throat]
>> and uh
And in general, the the moduli spaces of
sheaves on the projective space uh may
have several irreducible components.
They look like stuff like this.
>> [clears throat]
>> Uh
However, there are some uh well-behaved
cases. There are some cases for for the
Chern classes that we can say something
uh about these moduli spaces. In
principle, they are very
uh
very wild, but there are some cases in
which we can uh they are not so wild and
we can say something about them.
Here it should be C1 uh equals zero or
minus one.
It should be a minus one, sorry.
>> [snorts]
>> So,
first is the question of of
non-emptiness. Right? So, when
I'm always on range two here.
>> [clears throat]
>> So, if you fix these two numbers, the
first and the second turn class,
the third turn class cannot be too big.
Right? So, if it's larger than
something, we know that this moduli
space is always going to be empty.
Depending on the the the C1 and this
and the C2, if C3 is very large, this
moduli space is empty.
This is the existence.
Otherwise,
otherwise, if if the C3 is a smaller
than the this number here, then this is
non-empty provided
this parity condition is satisfied.
This parity condition is satisfied. I'm
implicitly assuming that C2 is positive
here. Even though C3 can be whatever, it
can be negative.
In the case of projective spaces, these
turn classes are just numbers.
Right? They can I can present them just
as numbers.
Uh
When
the third turn class, when this number
C3 is the largest possible,
then this moduli space is smooth,
irreducible. It has a known formula for
the dimension that I'm not going to
write here. Right? But, it's This is the
nicest possible situation. It's just one
piece that This
and you can classify these uh
uh
different decent enough
variety.
And also when
for these when C2 is
zero, C1 and C2 are zero.
Uh you can also say something about
these modular spaces.
Uh they are irreducible of a certain
dimension, and they are they are going
to be irreducible provided N is not
large enough. It's It's not too large,
right? Between
two and N. It has
to be any number between two and N.
But in general
modular spaces of sheaves on threefolds
obey what it's what became known as
Vakil's Murphy Law.
Every conceivable singularity
can be found on some Hilbert scheme or
modular space of sheaves. Right? So,
modular space of sheaves on threefolds
can be as bad as
you want. You can find any type of
singularity
in these modular spaces.
Uh
Sorry?
Uh
Yes.
No, yeah. Even if you start with a good
variety,
>> [clears throat]
>> you can on P3, you can find any
conceivable singularity
on
as as a module in some modular space of
stuff on that variety.
Now, Uh,
some cases in which the
the moduli space is not, you know, ultra
nice, but we can still tell something
about it.
Um,
so this situation, this this moduli
space is connected connected because for
for this case the rank two sheaves it's
not even know whether the moduli space
is connected. Can have a various
disconnected pieces.
And it has
two irreducible components. So let me
Let's see. So this is the case of uh
M
2 0 2
uh
So
four is the max.
So C3 max
here is equal to four. And in this case
we know that it's irreducible. So if we
fix
C2 equals two
and C1 equals zero
max is four. So there are two cases to
consider. The case C3 equals two
and also the case
2 0
2 0.
And then all the negative cases because
this C3 can be as negative as you want.
But
the these two cases
um
So this now has two components. One has
dimension 13 um
Yes. One has the has dimension 13 and
the other has dimension 17. And this guy
here has
three
components. One has dimension 13, the
other
um
17 and finally 21. They go by
by four.
>> [clears throat]
>> So they have two and three irreducible
components and we can tell what is the
genetic point in each of the components.
I can uh
we can tell uh what they look like.
Um now, if instead of uh C1 = 0, you put
C1 = -1 here,
then again, C3 max is again four. And
there are two cases that we studied.
These are two cases.
Uh
This again is going to have two
components of dimensions 11 and 15.
And this will have four components.
Uh
11
15
19 and the other I don't remember. I
should have checked. I want to say that
it's uh
17, but I'm not sure.
Um
And I drew it like that because I don't
know whether this component intersects
this other component. I only know that
it is intersecting one of these two.
Possibly the other one, but uh we only
proved for one of these two, but not for
uh the intersection with that one.
Uh
So, yeah, so these are the four
irreducible components and two
irreducible components
that we found.
Uh
Already in the this other space with
second chain class equals three
and third chain class zero has at least
11 irreducible components and we don't
know the total number.
Still haven't Still don't know
the total number of components. We found
11. We don't know if if we found them
all.
We also know that the number of
irreducible components grows to infinity
with C2.
Right, so the larger the second trend
class
the higher the number of components.
But we don't know how this growth
occurs.
If it's a polynomial, exponential,
logarithmic.
If it only grows, if it grows
diminishes and grows again, if it's more
whether or not this growth is mono is
monotone.
Right, so we don't know how how it
grows.
And
as I said, it's not even know in gen
known in general whether these moduli
spaces are are connected.
And
and the one of the new things we've been
trying to explore is what happens when
the third trend class is negative.
Right, this is also allowed. This is
also you know there are such shapes with
negative C3, but we haven't we just
started to study these moduli spaces
now.
Maybe
the last slide I'll
mention
one of the
the key
techniques or the key idea behind trying
to to understand these results.
This is done by the so-called sacred
springs. Sacred correspondence is a very
nice
link between vector bundles and curves
on P3.
It tells you that whenever you consider
a pair of
a bundle with a section
of a global section,
this is the same as considering a curve
or more generally a one-dimensional
scheme because it can be a curve with a
bunch of points.
They don't have to be pure dimensional.
So, it's it's a one-dimensional scheme
together with a section of this order
dualizing sheaf.
So, there is this correspondence.
And then you can
actually study the moduli space of pairs
of bundle with section.
And this has been studied. There's
literature about that. This is a
projective scheme. And what we found
recently is that for each irreducible
component of this
moduli space of pairs, you have rational
maps.
One map goes to
goes to the moduli space of sheaves,
which is this M here. And another
rational map takes you to the Hilbert
scheme of curves. Hilbert scheme is
something that classifies curves inside
P3.
And these are
And And you have these rational maps and
this is
This is one way to understand this. So,
for instance, in this particular
situation,
these
these irreducible component here, the
the sheaves in this irreducible
component, they will correspond to
twisted cubics
inside P3.
And I think it's twisted cubics.
Yeah, they should have degree three.
So, one is a twisted cubic and the and
this guy is a twisted cubic together
with a with a point.
>> [clears throat]
>> Is it a conic and a line or just a
conic? Conic and a conic and a point,
yeah.
I'm I'm confusing with this is the other
case, yeah. So, this is a conic
and this is a conic with a
with a floating point, yes. This is the
the
the the the planner in the
So, the
the
the sheaves, the generic sheaf in this
component here via this same
correspondence corresponds to plane
plane curves of degree two, conics.
And this component has to do with conics
with a with an added point somewhere.
This one is again conics,
conics and a point.
This is conic and two points.
And this
third component is
is something like a line
and a
or I guess it's a double line.
I I don't remember it.
Maybe
it's something like a double line with
some point as well.
Or the double line with one point or
But it
it comes from some non-reduced schemes.
So, it's understanding the the So,
And Leonardo's presentation will give
you more details on that, right? So,
that's what he will present next.
So,
the way to to study to understand these
irreducible components it's tying them
down to the irreducible components of
the Hilbert scheme.
And that's the
one of the the beautiful things about
this topic that it gives this relation
between the
uh
classification of sheaves and
classification of curves and how to go
from
uh one to the to the other.
Um
and that's all that I have
for today. Thank you very much.
>> [applause]
>> So, these conics are they're
intersecting at their common points.
No, they don't. Depending on the
So, yeah. So, so this is just one conic.
There's no other curve. Just uh
just a And two of them
Yeah, at this point may or may not. No,
these these are two different conics.
So, one uh family of sheaves correspond
to just one conic and the other family
of sheaves correspond to one conic plus
a point.
Right. So, these are uh two different
families.
Are there any general results about the
moduli space in dimension greater than
two?
As far as I know, only that uh they are
projective schemes.
Otherwise, they can be pretty much
anything.
And connectedness only for the Hilbert
scheme, which is the rank one case.
Yeah.
Are there non-commutative analogues of
the rank of
the
LPS directions and Um
I think there should be non-commutative
analogs of moduli spaces,
so for instance, people do
in super geometry
it's the kind of non-commutative
geometry where the non-commutative
non-commutativity is very mild, just has
these Z2
grading on the algebra.
And I know that in the case of super
geometry,
there is a theory of moduli spaces
for bundles on super manifolds.
So it's possible that this is
viable also for non-commutative
non-commutative as well.
It would be classifying all modules
and and this has also been done for
associative algebras.
Classification of
of a modules over associative algebras,
they also admit moduli spaces.
So
it's possible that there is something in
the
more generally in non-commutative
geometry.
Thanks everyone.
I think we take a little break before
Take a 5-minute Yeah, 5-minute break.
>> [cough]
>> Before the last [clears throat] bit.
Okay.
Yeah.
Okay.
>> [snorts]
[snorts]
>> Uh Professor, please share the screen on
Zoom. Okay, sorry. Thank you.
Yeah, I'm
Whereabouts?
Um there should be maybe in the top left
a little box with Zoom.
Uh the Z in there?
Uh hold on.
Okay, and then you have gone view
on the top.
Uh top left, you've got view.
And [snorts]
uh full screen. Here.
Um
No, it's not there.
You have to okay something here.
I don't know why it's not giving you
full screen. Okay, I'll be there in a
second. Hold on. Okay.
Oh, here it is. Here it goes.
We managed to do it here. You don't have
to come down. Thanks.
Uh so, first I want [clears throat] to
thank you to invite this presentation
and Marcus to invite me to be at here at
I speak.
So, I going to talk about the moduli
space of torsion free sheaves
with quasi maximal C3, where is C3 is
the third Chern class that Marcus talked
before.
This is [clears throat] a joint work
with Charles Almeida, my co-advisor, and
Marcus, my advisor.
And first we going to talk about the
historical background of this
how developed this study of
moduli spaces,
some basic definitions,
the moduli space it corresponds to the
question that Marcus talked before,
and the moduli space we are going to
show you how it is, and some next steps.
So, first we going to denote by M C1,
C2, and C3
the Gieseker-Maruyama moduli scheme that
parametrize semistable rank two sheaves.
So, Marcus
has a R before the C1 for the rank. Here
we going to talk about just rank two
sheaves
>> [clears throat]
>> on projective space P3.
C CI denotes the Chern classes and we
can consider this subscheme that's
denoted by R and C1, C2, and C3. This
one are going to be just the stable
reflexive sheaves. So, the double dual
the sheaves is of mark is double dual.
So,
this is a projective scheme.
And for Maruyama that there are the
Maruyama bounds that he can be very bad.
And these two papers say there is some
examples how this happen.
The moduli space have reducible
components of different dimensions.
So, we can fix the Chern class
as minus one. We have two options for
sheaves on P3 of rank two is zero or
minus one. So, we're going to fix minus
one.
And we have that C2 and C3 have some
possibilities. For example, C2 need to
be positive.
And then we have this condition on
parity of C3.
And this bound for C3 it's always
um
less or equal than C2 squared to be a
stable reflexive to be a stable sheaf.
We can have sheaves with more larger but
it's not going to be stable
or semi-stable.
So, first Hartshorne proved that for
when we have C1 equals minus one,
all these stable reflexive sheaves are
going to have a third Chern class less
or equal than the second Chern class
squared.
And
5 years after O'Grady and Spindler
proved that this bound remains true for
free sheaves. So, Hartshorne proves just
for reflexive and 5 years
after we have this bound for
torsion-free sheaves.
And Hartshorne computes the moduli
spaces when we have the maximal C3
for reflexive sheaves. And O'Grady and
Spindler computes the moduli space with
maximal C3 for torsion-free sheaves with
C2 greater or equal than 6.
And
recently we have this paper by Schmidt
that with different techniques he
computed the moduli space for all C2
with the third Chern class
being the maximal.
But when we talk about reflexive
sheaves,
they are that
things that I
that the means that I say before that uh
uh
>> [clears throat]
>> C3
is less or equal than C2 squared, but
there are some
C3 we don't have stable reflexive
sheaves. For example, Schenk studied
this for fixing C1
equals minus one and C2 greater than
zero, we have that for this range we
don't have a stable reflexive sheaves.
Some years after uh Miro-Roig
extended this result. So, if you fix
this speed as this number,
so for C2 greater or equal than four,
we're going to have a range that is no
stable reflexive sheaves.
So,
>> [clears throat]
>> until now we have those moduli spaces.
For
uh C2 C3 greater than C2 squared we
don't have stable sheaves and for
reflexive sheaves there is some gap in
C3 that is no stable reflexive sheaves.
But what about uh the torsion-free
sheaves?
So, there is a paper
uh where Almeida, Jardim, and Kumar
construct a possibly new family of
irreducible components for certain
moduli spaces.
This is this big theorem and we can
say like that. We start with a sheaf,
reflexive sheaf
with C1, C2,
and C3,
and you take a subjective map to O.
S
>> [applause]
>> where S is just S points on the tree.
So, you can take the kernel of this map.
I going to put
a
prime here. So, we can take this kernel
of this map and we going to have a sheet
here that we going to have Chern class
C1, C2,
C3 - 2S. What is S is the amount of
weights we have here.
So, they construct this new family. So,
if you have
a
a sheet in this reflexive the moduli
space of this reflexive sheets, we can
construct some sheet in another moduli
space and this one going to be a
torsion-free sheet.
So, but [clears throat] the it's not
just a stable reflexive sheet, we need
to be a smooth point in the moduli
space.
So, if you have a smooth point in our
moduli space of stable reflexive sheets,
we can do a transformation by a by
points and have a sheet that is a
torsion-free with this Chern classes.
So, this this theorem is like that. So,
for any reducible component of our
>> [clears throat]
>> the reflexive sheets, we can construct a
new
new sheets on
this new Chern class.
And we have the dimension of this
component.
It's going to be 8 * n - 3 + 2 e + 4S on
n and n, e, and S are
a relation as with the Chern classes of
the first weight.
>> [clears throat]
>> In particular, if you the moduli space
of the reflexive sheets is reducible, so
the this component is going to be
reducible, too.
>> [clears throat]
>> So, now I can go to some basic
definitions. I'm going to start with
what is a stable sheaf?
To do that, first uh we remember what is
the Euler characteristic of a sheaf.
So, for this we're going to take the
homology, the alternate sum of the
dimension of the homology of the
sheaves.
So,
this [clears throat] homology stops in
the dimension of the our variety. In our
case, it's three.
And we can define the Hilbert polynomial
of a sheaf.
So, we're going to take
uh this polynomial as just take the
Euler characteristic of the sheaf
twisted by the integer.
And this is going to be a polynomial
this function going to be a polynomial,
and we can define the reduced Hilbert
polynomial. We're going to take the
Hilbert polynomial and divide by the
rank of the sheaf.
>> [clears throat]
>> So, what's going to be a stable sheaf?
Uh
a torsion-free sheaf is going to be
Gieseker-semistable
if for any proper subsheaf you have
the polynomial the reduced Hilbert
polynomial of the uh subsheaf is lesser
or equal than the
reduced polynomial of the original
sheaf.
And if this inequality is strict, we're
going to say that E is just stable.
There is another uh definition of
uh stability that is the slope
stability.
And we start with a torsion-free sheaf,
we take the degree of the sheaf is
invariant, and the rank of the sheaf,
and when you take this
fraction, we have uh what we call the
slope of the sheaf.
So, they're going to be the same as the
Gieseker-stable. We're going to say that
a sheaf is stable if for semistable if
for all subsheaves, we have a
rank less or equal than our sheaf or
zero, [clears throat]
we have that the mu-stability the mu of
F is less less or equal than mu of E.
And it going to be stable if the
inequality is strict.
We have these implications. So,
mu-stable implies stable
implies semi-stable, that implies
mu-semi-stable.
But we are working with rank two sheaves
with the first Chern class minus one.
In this case, all the notions of
stabilities are equal.
So, mu-stable it's equal to
um
Gieseker stable and all semi-stable
sheaves is in fact a stable sheaf.
Now, I going to talk about a little
about the moduli stack of polystable
>> [clears throat]
>> This can be found in this paper.
So,
here we put the X with a smooth
projective fold.
But we can all the things I do is just
for P3.
So, first thing we can define is a
coherent pair.
Going to be a sheaf and a global section
of the sheaf.
Possible zero.
It's possible or you can take this
global section as zero.
So, we going to say that this pair is
pure if the sheaf is pure and S is a
non-trivial section.
And we can define now
stability notion on this pair.
So, we are going to take the
If you forget about the epsilon and the
delta, it's just a reduced Hilbert
polynomial. But you are perturbing by
this delta. Because if the section is
non-zero, so we have that
epsilon is one, so you are perturbing
our
reduced Hilbert polynomial and the
section is non-zero, you don't do
nothing, Just take the same Hubert pair,
reduce it to Hubert pair. No, no.
And this is the paint of the delta.
Delta is a just a polynomial
that is positive. Here are ordinary
polynomials if the voting is
one or other.
So,
>> [clears throat]
>> this is what we call the
delta stability.
So, when I say that it's stable,
semi-stable,
if for any proper super pair, we have
this inequality here. And if it's
strict, we call that it's delta stable.
>> [clears throat and cough]
>> Uh
Okay, we have that for example in our
case in a Gieseker
with rank two and first Chern class
minus one, we don't have any proper
semi-stable sheaf.
But here we can have.
And when we have, it's depends on the
delta. And if the delta
if for this delta, there is semi-stable
sheaf, proper semi-stable sheaf, we call
this delta a wall.
>> [clears throat]
>> And we can define the collapsing wall.
What What we call collapsing? Because if
you take delta as this polynomial, every
pair is going to be unstable.
So,
we have this
>> [clears throat]
>> delta stability and for every delta
greater than this one, everyone is
unstable.
And we have another definition
that's now depends of the our section.
So, if I have a pure coherent coherent
pair, it's called saturated if the
co-kernel of the
monomorphism of the section is torsion
free.
So, you have a pair, look to the section
which is this induced a monomorphism
from O to the my sheaf. And if the
co-kernel of this monography's is
torsion free, we call the pair
saturated.
>> [clears throat]
>> And we can define now a second wall.
This wall is the maximum possible wall
that is not greater than the collapsing
wall.
So, what happens is between these two
walls, this new wall and the the
collapsing wall,
all the sheaves
are torsion free are saturated.
So,
>> [clears throat]
>> we have
This line
is our delta.
And this is
the collapsing wall. So, after this,
every pair is unstable and we have
this [clears throat] one.
So, everyone here is saturated.
And you know how big is the
this chamber. Chamber's thing between
two walls.
And we can define So, we define the last
wall, the second last wall. Now, we're
going to to the first wall.
Because we start with with no
positive polynomials.
So, we we have
zero here.
And this first chamber
and we can have chambers here.
And this first chamber we're going to be
the
>> [clears throat]
>> a chamber that every stable Oh, sorry.
Define the very stable first.
We're going to say that a
pair is very stable if the is delta
stable as a pair and my sheaf is is
curve stable.
What happens [clears throat] is between
this chamber, everyone that um
Oh, sorry.
Every sheaf can that is stable a
section, we're going to build up the
stable.
So, until now we have this structure,
this line of polynomials, and this wall
in chambers here.
And that is this theorem that when we
consider
the space of pairs
here,
we can have
these two maps.
Uh
when we choose delta between these two
walls in this chamber, we're going to
have a map from the from the space of
pairs to the Hilbert scheme.
And when I choose delta between zero and
this first wall, we're going to have our
map
from the space of pairs to the moduli
space of sheaves.
And [clears throat] we know uh how is
this map.
We know the fibers.
So, now what we're going to do?
Uh
If you don't have any wall,
because just the collapsing wall uh it's
always exist, but the other ones maybe
or not it exist.
If you don't have any walls, so these
chambers coincide. So, we're going to
have these two maps at the same time.
And how you know this uh the Hilbert
scheme, if you know the Hilbert scheme,
we can use this map to
have information about the geometry and
topology of this this curve.
I don't know what happened.
I don't know.
Oh, okay.
Uh
so, you can use the geometric and
topological information of the Hilbert
scheme. Sorry, I'm going to back.
Of the Hilbert scheme to understand
better what is the disk curve moduli
space.
So, we want to see how this work in our
case.
The first one we're going to see that
every pair with a C2 greater or equal
than three
the torsion the
torsion-free sheaf with quasi-maximal
Chern class have a non-zero section.
I put [clears throat]
greater or equal than three, but the
demonstration I have here is just for
the C2 equals three. Sorry.
Uh to do this demonstration, first we
need to recall what is the spectrum of
the sheaf.
It's going to be a collection of
numbers,
integers in fact,
that the cohomology of our sheaf twisted
by some number
is the base of these numbers.
So,
the case we're testing here is the
the second one.
Uh
Okay.
So, the second cohomology of our sheaf E
depends of the first cohomology of the
structural sheaf of P1
twisted by the spectrum plus the first
twist here plus one.
So, you can use this this cohomology to
compute the cohomology of E if you know
what is the spectrum.
And there is more about the spectrum.
For
if you look it's just uh
some numbers. We don't have how many
numbers, for example, the first case
with this theorem
and any condition about the number.
>> [snorts]
>> Yeah, yeah, you
to construct this number you restrict to
a plane first and look to the plane the
sheaf in the plane.
I do know.
Okay, I don't know. I
It doesn't have to do with the splitting
on
two ones.
No, it's not because if you one for a
two, you would get just two ones.
No, yeah,
that's the general genetic splitting
type.
>> Yeah.
For a rank r.
This this is the spectrum. The K I
AM is just a
the generic splitting type. So, the
spectrum is this number K
and [snorts] this number K that will
happen is in the homology.
What does M mean? M is The the
amount of numbers we have in the
spectrum.
They are going to be the C2.
Okay. So, the number of in the spectrum
is C2.
Uh
okay. So, you have this uh
proposition that say if you have the
spectrum of the sheaf
and you take uh
negative number
so, every number between this negative
number and minus one has to be in the
spectrum
too.
So, we have a condition to say how the
sheaf how
uh how many how many numbers do you are
right now is always the C2, but now we
can say more about how negative you can
have a number in the spectrum.
So, going back to our lemma
first we have that uh Euler
characteristic of our sheaf is going to
be that and you can use Heisenberg to my
heart to calculate the Euler
characteristic of the sheaf and it's
going to be one.
As the sheaf is stable, we have the
third homology is zero.
So, we have that the first the global
sections minus the first homology plus
the second one equals one.
If the second homology of our sheaf is
going zero,
so we have that the our sheaf admits at
least one global section.
And we are doing that by analyzing the
spectrum. So, when you look to the
spectrum of the sheaf, is that firm?
Uh
so, for
the H2 not be zero,
we have that minus four is going to be
in the spectrum
because of the homology of P1.
But, by our
uh
proposition we see before, if minus four
in the spectrum, so I have minus three,
minus two, minus one.
But, we have only M uh C2 numbers in the
spectrum.
So, we can have minus four in the
spectrum. So, this homology need to be
zero.
And as you as you see, if it's zero, so
our sheaf admits a global section.
>> [clears throat]
>> Uh we have this proposition that if I if
I have a torsion-free sheaf with rank R
and have a section, no new section,
so for every
non-trivial section, if our sheaf is
stable, so the pair is going to be uh
delta stable.
And we see that very stable is delta
stable plus the sheaf is going to be
stable, so our pair is going to be very
stable.
Uh
the [clears throat] other lemma we need
is prove that okay, we have a global
section,
but uh this pair is saturated. And we
can prove that in fact the Z
the Z is saturated um
and you have how we know what is the
the co-kernel of this map.
And as I I say before, we have to see if
there is more another ones or just the
classical one.
So, a possible one going to be a sub
object that have the same stability
because of all this when I have a
strictly a proper semi-stable pair.
So, I need to see if it's possible in
that case I have a proper semi-stable
pair for any doubt.
So, we going to be defined by a sheaf
and a section. [clears throat]
And the first thing we can look is that
if the this sheaf don't have any global
section, so the delta Hilbert polynomial
coincides with the Hilbert polynomial.
So, we going to have the same Hilbert
polynomial for this case if they don't
have a global section.
As our sheaf
we state start with a pair.
Our sheaf L is going to be a sub sheaf
of our
initial sheaf E twisted by one.
But E is stable.
So, I know that
the Hilbert polynomial of L is going to
be strictly less than the Hilbert
polynomial of delta of of E.
So, we have this inequality here.
But you can consider the following
diagram.
We start with E.
Oh, okay.
And take the double dual.
And we going to have just a structural
sheaf of a point here.
And here we have L as
a sub sheaf.
As a double dual going to be the
structural sheaf O. So, you have this
map O to the double dual to the
structural of a point.
We take the composition. We have this
composition going to be or new or no
new.
If this composition is zero, so we have
that L is going to be just uh, this is
sheaf.
If it's no move, so it's subjective and
we're going to see going to have a
rank one sheaf into the structural sheaf
to the structural of a point. So, L
going to be the ideal sheaf of this
point.
So, the only three sweepers we have for
this
uh, our our pair is the structural and
the zero and the loop section.
The structural and the section that is
the collapsing wall and the ideal sheaf
the ideal sheaf of this point and don't
have any global section, so zero.
As E is a stable
we have that
uh, O and IP and the ideal of the the
point not going to stabilize our sheaf
because our sheaf is stable this two are
same our sweeper our sub sheaves.
As he is a stable not stabilizer E.
Okay?
So, [clears throat]
uh
we have don't have walls in our case.
So, at the same time we have this
morphism to the Gieseker
the Gieseker moduli space and this one
to the Hilbert
uh, Hilbert scheme.
>> [clears throat]
>> That's
Uh, but what Hilbert scheme is that?
That one. We can calculate it.
Uh, you know that
going to be this section this uh, exact
sequence
the section and so the kernel is ideals
of something. We can calculate with the
Chern class.
And looking to the structural the of the
sub scheme sub scheme Y we can calculate
it the Hilbert polynomial and we have
this three three times T plus one.
And this system scheme you know what
what is.
So is the union of two no singular
rational varieties 80 and 8 prime.
Where 80 is a twisted cubics.
And 8 prime is
planar cubics
plus
a point.
And have a intersection. So it's
connected.
>> [clears throat]
>> So if
if this map going to be subjective we
can use
the properties of the Hilbert scheme to
conclude things about the his command of
spaces.
So this is the position. We have
if we take X1 of the ideal of the chief
by O
twisted by minus one, this is no loop.
So we have excitations. So we're going
to have our stable chief
in the
space of first.
And [clears throat] we can complete that
and prove that this in fact no loop.
And now we have
this map
is subjective. We know this one have two
components. So this one have two
components. And what is this components?
One is going to be chief associated with
twisted cubics. And another one going to
be a planar cubic plus a point.
This one is about this T component I
said before.
And this one going to be the reflexive
chiefs.
Where you know what is. Is a reducible
rational of dimension 19 and generically
reduced.
So [clears throat] you know what is the
moduli spaces with different classes has
two irreducible components.
The component of the reflexive is
associated to the twisted cubicity. and
the thick component that is associated
to the final cubic plus one point.
And do you know the dimensions of
uh each component?
This is the references.
And thank you.
>> [applause]
>> I think that you have to, you know, read
your slides very carefully because they
were they went a bit fast. Oh, sorry. I
kind of anxious.
Just one comment. If you go back a
couple of slides. Okay. Which one?
The the one that has the two maps.
Yeah, this one.
Uh except that here they are not right
since you showed that there are no
walls. Yeah. This is not uh these are
not rational maps. These are really
morphisms. Oh, okay.
>> actual maps between Okay.
Thank you.
>> [clears throat]
>> So, what's this result of Schmidt?
Uh
Okay.
I go back to
where I
say.
Yeah.
Uh
because Hartshorne uh do all these
things for reflexive sheaves. So, 5
years after, Oprea and Klee proved that
uh the bound is the same and calculated
the
>> [clears throat]
>> the moduli space for the maximal ones
with C2 greater or equal than 6.
Schmidt uh made a different uh approach
and he proved for every C2 in a
different very different way of the
connected spinner.
Yeah, it shows that the moduli space is
irreducible and smooth. It's smooth,
irreducible, smooth, and say what is
exactly. And it looks like a
Grassmannian vibration over something.
>> Yeah. It gives a very complete model.
And this is the nicest possible space.
>> Yeah.
This is the nicest possible.
Yeah, and then this case
for C2 equal three, we have two
components, but
for C2 greater or equal than four, so
there are no reflexive sheaves. So, the
moduli space going to be just this C
component that is a planar curve plus a
point.
It's a nice one, too.
This is what Leonardo is planning to say
tomorrow.
>> Yeah.
Explain the case C2
and larger than six.
And any of this can it be extended to
say P1 cross P2 like
I don't know. Has there any study been
done for moduli space on
P1 cross P2?
Say P1 cross P2 or
Not that I know. Or blow up formula?
The
the
limitation is understanding the Hilbert
scheme of curves
on whatever.
Right? Because this result is
like
any any threefold
you have these rational maps and so on.
Yeah. But then you have to know the
Hilbert scheme
of curves on
the what type of on X. So, we have two
of these points.
>> to apply this this idea.
Okay, I think it'd be nice to have
examples to apply this thing to the
water two points.
I actually think the intersection theory
just on the surface P2 is hard. I don't
know if you
I mean, intersection numbers, for
example, on the Hilbert scheme of points
on P2
could be crazy. It's not
So, P2 is not easy as a surface to
apply, you know, you know?
Yeah, and there is a kind of luck
because
there is a paper that say about Hilbert
scheme with a fixed degree and the genus
of the subscheme and say for a range is
irreducible. And we know what is. It's
just a planar curve with a strict
So, this help a lot.
And to compute topology intersection on
these on these
Yeah, even if you give me a let's just
just one component.
It's out of my hands.
Out of my reach.
It's not computable.
I understand [applause] that
It's a
Marks.