All Questions
Tagged with enumerative-geometry gromov-witten-theory
26 questions
3
votes
0
answers
165
views
Enumerative or Gromov-Witten invariants from derived category of coherent sheaves
Let $X$ be a smooth projective toric Fano surface over $\mathbb{C}$. Suppose I have a nice presentation of $D^b_{Coh}(X)$ given by a full, strong exceptional collection $\mathcal{E} = \{E_i\}_{i\in I}$...
- 511
11
votes
2
answers
1k
views
Simple examples of Gromov-Witten invariants not being enumerative
I understand why Gromov-Witten invariants in general are not enumerative, so it's not necessary to explain this. However to test something I'm working on, I'm looking for examples of concrete ...
- 111
2
votes
1
answer
150
views
Counting maximally tangent conics relative to a cubic
Is it possible to count the number of conics in $\mathbb{P}^2$ that are fully tangent at one point to a given (generic) cubic curve using basic intersection theory calculations?
The corresponding ...
4
votes
2
answers
584
views
Explicitly computing Donaldson-Thomas invariants (of CY 3-folds)
I am interested in the explicit computation of generating functions of rank 1 and higher rank Donaldson-Thomas (DT) invariants. In particular, I am interested in DT invariants of K3 fibered Calabi-Yau ...
user35360
2
votes
1
answer
618
views
Reference request for Gromov-Witten Invariants of non compact manifolds
The title of my question essentially explains what I am looking for, but let elaborate a bit, to put it in a more specific context.
There are quite a few papers, where the authors compute Gromov-...
- 3,245
5
votes
1
answer
207
views
Curve-counting with fixed source
Suppose I fix a smooth projective curve $C$ of positive genus, and I have a smooth projective variety $X$. Do standard tools from GW theory (or any curve-counting theory for that matter) allow me to ...
- 735
2
votes
0
answers
99
views
Diagonal operator and infinite wedge space formalism
Let $\bigwedge^{\infty /2}V$ denote semiinfinte wedge space. The followin article section 2 gives a good description about the space and the operator on it.
https://arxiv.org/pdf/math/0207233.pdf
...
- 685
4
votes
0
answers
212
views
How does one obtain the formula for the number of genus one curves in P^2 using Getzler's relation?
I am trying to get the formula for the number of degree $n$ genus one curves in $\mathbb{P}^2$ passing through 3n generic points, by following the procedure in Getzler's paper
https://arxiv.org/pdf/...
- 3,245
1
vote
0
answers
241
views
De Jonquières formula vs. Relative GW invariants
Background. Let $C$ be a smooth projective curve of genus $g$. Denote by $C^d$ its d-th symmetric product and by $G^r_d(C)$ its associated variety
of linear series of type $\mathfrak{g}^r_d$, i.e.
$$ ...
- 1,981
11
votes
1
answer
708
views
Is there a tropical geometric proof for counting genus g curves in any n dimensional projective space?
Consider the following question: Let $X$ be a compact complex manifold
and $\beta \in H_2(X, \mathbb{Z})$ a fixed homology class. Let
$\mu_1, \mu_2, \ldots, \mu_k$ denote certain generic ...
- 3,245
1
vote
0
answers
266
views
How does one define Moduli spaces in Symplectic Geometry and naively interpret higher genus GW Invariants?
This is a very basic question about the definition of Moduli space of maps.
My reason for asking this question is because I haven't actually seen this
definition explicitly given anywhere, and hence ...
- 3,245
7
votes
1
answer
706
views
Are genus zero Gromov Witten Invariants on Del-Pezzo surfaces enumerative?
Let $X_k$ be $\mathbb{P}^2$ blown up at $k$ points (where $k$ is $0$ to
$8$). Let $\beta \in H_2(X_k, \mathbb{Z})$ be the homology class given by
$$ \beta := n L + m_1 E_1 + \ldots + m_k E_k $$
...
- 3,245
4
votes
1
answer
305
views
What are the indecomposable classes on a del-Pezzo surface?
Let $X_k$ be $\mathbb{P}^2$ blown up at $k$ points (where $k$ is $0$ to $8$).
Let $\beta \in H_2(X_k, \mathbb{Z}) $ be a homology class given by
$$ \beta := n L + m_1 E_1 + \ldots + m_k E_k $$
...
- 3,245
2
votes
0
answers
162
views
Degree 2 curves on a degree d hypersurface in P^(2d+2)/3
One of the foundations of Gromov-Witten theory is the use (due to Kontsevich I think) of localization to calculate the number of degree $n$ curves on a general quintic 3-fold. When calculating the ...
- 265
2
votes
1
answer
235
views
Is there a formula for the number of rational cuspidal curves in surfaces other than P^2?
Let $M$ be a two dimensional compact complex manifold and $A \in H_2(M, \mathbb{Z})$
a fixed homology class. Define a rational curve in $M$ to be $\textit{1-cuspidal}$ if the singularities of the ...
- 3,245
9
votes
0
answers
860
views
Question on Ionel and Parker's paper: Relative Gromov Witten Invariants
In defining Gromov-Witten invariants using symplectic geometry, most of the trouble is to achieve transversality for moduli spaces of pseudo-holomorphic curves which are multiple covers of simple ones....
- 1,331
3
votes
0
answers
270
views
holomorphic embeddings of the sphere into the quintic in degree 2
Is there an explicit way of
classifying (with regard to their compatibiliy with $\Omega_+$ or $\Omega_-,$ see below)
the various families of
equivariant holomorphic embeddings from $\mathbb{CP}^1$ to ...
5
votes
1
answer
345
views
Looking for a reference (on GW invariants of quintic)
1) Apparently, physicist can calculate the GW invariants of quintic CY 3-fold up to genus 51.
I am looking for a reference that has a table of these number for some low degrees (say up to degree 5) ...
3
votes
2
answers
743
views
Conics in the quadric line complex
Hello, I apologize in advance if this question is misguided somehow, since my algebraic geometry is pretty shaky.
I am wondering if there is a way to understand all the conics in a generic quadric ...
- 1,129
3
votes
1
answer
1k
views
Is P^2 important in Kontsevich's recursion formula?
There is a famous recursion formula by Kontsevich to find the number of
genus zero degree $d$ curves in $\mathbb{CP}^2$ through $3d-1$ points.
My question is the following: Let $S$ be a complex ...
- 3,245
2
votes
0
answers
174
views
Are there ways to make low degree checks for enumerative formulas except for curves in CP^2?
This is a concrete question in Enumerative geometry. Let $S$ be a compact
complex surface and $L\rightarrow S$ a holomorphic line bundle. Let
$$ \delta_d = \text{dim}~ \mathbb{P}(H^0(S,L^d)) $$
...
- 3,245
13
votes
1
answer
2k
views
Who streamlined Kontsevich's count of rational curves?
Let $N_d$ denote the number of rational curves in $\mathbf P^2$ passing through $3d-1$ points in general position. Maxim Kontsevich discovered a famous recursion for these numbers:
$$ N_d = \sum_{k+l =...
- 40.2k
1
vote
1
answer
500
views
Does any one understand the details of M Kazarian's work in enumerative geometry of $\mathbb{C}\mathbb{P}^2$ ?
I wanted to know if anyone understood the details of the paper
"Multisingularities, cobordisms, and enumerative geometry" available at the site
http://www.mi.ras.ru/~kazarian/.
In particular does ...
- 3,245
5
votes
2
answers
647
views
Is anything known about the enumeration of degree d, genus g curves in CP^2 where g >1 ?
I wanted to know if there is something analogous to Kontsevich's recursion formula for
enumeration of genus zero curves in $\mathbb{C}\mathbb{P}^2$, for higher genus curves.
There is a
similar ...
- 3,245
2
votes
1
answer
643
views
Computing 3 points Gromov-Witten invariants of the Grassmannian
This is from an exercise in Koch, Vainsencher - An invitation to quamtum cohomology.
Background
The exercise asks to compute the 3-points Gromov-Witten invariants of the Grassmannian $G = \mathop{Gr}...
- 14.7k
23
votes
7
answers
2k
views
Higher-dimensional Gromov-Witten theories
A basic set-up in modern enumerative geometry is that you have some object $X$ (say, a "nice" stack, for whatever definition of "nice" you need) and then you want to "count&...
- 16k