Questions tagged [enumerative-geometry]
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73
questions
3
votes
1answer
144 views
Configuration of points on a plane curve
Let $C\subset\mathbb{P}^2$ be a smooth plane curve of degree six. On $C$ there are $21$ points given as the intersection points of two lines choosen among a set of seven lines. More precisely there ...
5
votes
0answers
153 views
How to compute the class defined by intersection with a square?
$\DeclareMathOperator\Gr{Gr}$Let $\Gr(k,n+k)$ (of course, one can do also for $\Gr(k,\infty)$) be the complex Grassmannian of $k$-planes in $n+k$-dimensional linear space.
It is well-known that ...
2
votes
0answers
134 views
Quadrics tangent to lines
I think that the following must be a basic question in enumerative geometry.
Take a line $L\subset\mathbb{P}^3$. The quadric surfaces in $\mathbb{P}^3$ that are tangent to $L$ are parametrized by a ...
8
votes
1answer
408 views
Interpretation of “27” lines for cubic surface with rational double points
It is well known that a smooth cubic surface has $27$ distinct lines. Explicitly, if we choose a planar representation, i.e., blowup $\mathbb P^2$ at $6$ general points $p_1,...,p_6$, the $27$ lines ...
0
votes
0answers
87 views
Counting quadratic curves in $\mathbb P^1 \times \mathbb P^1$ passing through seven points in general position
Let $p_1,\dots,p_7 \in \mathbb P^1 \times \mathbb P^1$ be 7 points in general position.
What is the number of maps $F=(F_0,F_1):\mathbb P^1 \to \mathbb P^1 \times \mathbb P^1$ modulo domain ...
6
votes
2answers
633 views
Conjectures and open problems in representation theory [closed]
Are there very famous open problems or conjectures in representation theory, or in enumerative geometry, like the volume conjecture in topology?
4
votes
1answer
277 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 ...
2
votes
1answer
345 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-...
1
vote
0answers
138 views
Log-canonical bundle of a smooth curve with marked points
I am not sure if this question is appropriate for this site, but here it goes. I am not a geometer, so I am not familiar with notation in the area.
I am interested in the moduli space of $r$-spin ...
2
votes
0answers
165 views
Number of degree d curves passing through d points in the projective plane over a finite field
Let the base field be a finite field $\mathbb F_q$ and fix $d$ rational points that lie on a line in $\mathbb P^2$. Suppose $d$ is a large number (about the order of $q^{\alpha}$ for $\alpha$ some ...
3
votes
0answers
80 views
Galois group for triply tangent planes of generic algebraic surface in $\mathbb{P}^3$
Background: The paper of Joe Harris asserts that for $d\geq 5$, the Galois group of $d(d-2)(d^2-9)/2$ bitangents of generic plane algebraic curve is the full symmetry group. I am wondering whether ...
1
vote
0answers
23 views
Extension of definition of Holonomic closure
My question is about finding the annihilator of a series. Let me begin with what is known and then ask my question. Let $s_d(\frac{q_1}{h},\ldots )$ denote schur function for partition $\lambda =[d]$ ...
5
votes
1answer
181 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 ...
2
votes
0answers
77 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
...
10
votes
1answer
427 views
Proving Positivity for Schubert Calculus
In study of the cohomology ring of the Grassmannians, which is usually known as Schubert calculus, one usually deals with a distinguished basis known as the Schubert basis $\{\sigma_\lambda\}$. One of ...
4
votes
0answers
164 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/...
4
votes
0answers
93 views
Closed form for integer series from enumerative geometry problem?
Is there a closed form for the following integer sequence:
$$
1,6,145,8806,830622,100317140,14342519633,2325250316950,...
$$
This is the degree of the $2n$-th power of the Schubert class $\sigma_{2,...
6
votes
2answers
391 views
Some Elementary Schubert Calculus Calculations
Here are some simple geometry problems I am unable to resolve to my satisfaction. I asked the question on Math Stack (https://math.stackexchange.com/questions/2713754/a-problem-in-elementary-...
1
vote
1answer
225 views
Hilbert scheme of points and passing curves
It is well known that through five points on a projective plane you can pass a conic.
Q. What happens when points collide ?
More precisely: if I consider a more simple question of two points and ...
3
votes
2answers
199 views
how many bitangents on this hypotrochoid?
After playing with spirograph, a bit I realized all these curves I'm drawing should be an algebraic curve and it's birational equivalent to a $\mathbb{P}^1$. In the example below, I have a six-sided ...
20
votes
4answers
2k views
What is an “integrable hierarchy”? (to a mathematician)
This is one of those "what is an $X$?" questions so let me apologize in advance.
By now I have already encountered the phrase "integrable hierarchy" in mathematical contexts (in particular the so ...
5
votes
1answer
205 views
Largest number of points one can pick in finite projective space without getting three on a line
Consider the projectivization $\mathbb P\mathbb F_p^n$ of $\mathbb F_p^n$. How large a set $B \subseteq \mathbb P \mathbb F_p^n$ can I pick so that no three points of $B$ lie on the same line?
2
votes
1answer
143 views
Bounding number of $k$-nearest neighbor sets in $\mathbb{R}^d$
Suppose that $\mathcal{X} \subseteq \mathbb{R}^d$ is compact.
Let there be $n$ distinct points $X = \{ x_1,...,x_n \} \subseteq \mathcal{X}$ and $k = \lfloor n^\alpha \rfloor$ where $0 < \alpha &...
11
votes
0answers
299 views
3264 rational conics
It's a classical fact that there are 3264 plane conics tangent to 5 general conics, over $\mathbb{C}$ [1]. It was also shown that the 3264 can be defined over the reals [2] or [3]. More precisely, ...
1
vote
0answers
197 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.
$$ ...
2
votes
0answers
440 views
Proofs that the Plücker relations generate the ideal of the Grassmannian
Some context: The $(k,n)$-Grassmannian is the set of $k$-dimensional subspaces of an $n$-dimensional vector space $V$. It can be realized as a projective variety via the Plücker embedding, and the set ...
4
votes
0answers
96 views
Random polyominoes containing $2\times2$ squares
The construction quoted in the question "How to sample a uniform random polyomino?" claims to produce a "uniform random polyomino". But apart from the mentioned possibility of getting stuck, it also ...
2
votes
0answers
142 views
Enumerating the number of degree d curves tangent to a planar conic
This question is based on a special case of the Coparaso Harris formula, as described in Counting curves on rational surfaces - R. Vakil.
Let $E$ be a non-singular planar conic.
Then every degree $d$...
2
votes
0answers
185 views
Schubert Calculus for the Full Flags
Almost all introductory texts on Schubert calculus discuss the Grassmannian case only. Does there exist a nice discussion of the full flag manifold case $SU(N)/T^{N-1}$? A low dimensional example ...
2
votes
0answers
120 views
Listing all Lattice Points in a Box
Let $B := [-1,1]^n$ be an $n$-dimensional box. Moreover, let $v_1,\ldots,v_n \in \mathbb{R}^n$ form a basis of $\mathbb{R}^n$, where the entries of the $v_i$ are explicitly irrational. We can assume ...
10
votes
1answer
588 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 ...
1
vote
0answers
247 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 ...
6
votes
1answer
559 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 $$
...
3
votes
1answer
277 views
What are the indecomposable classes on a del-Pezzo surface?
Let $X_k$ be $\mathbb{P}^2$ blown up at $k$ points (where $k$ is $0$ to $8$).
Let $\beta \in H_2(X_k, \mathbb{Z}) $ be a homology class given by
$$ \beta := n L + m_1 E_1 + \ldots + m_k E_k $$
...
2
votes
0answers
147 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 ...
2
votes
1answer
220 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 ...
7
votes
0answers
155 views
How big can a family of pairwise intesecting affine spaces be?
I apologize if this question might seem to be a bit too elementary.
Let $\mathbb{P}^n$ be the projective space over $k$ - an algebraically closed field of characteristic 0. Let $1\leq l\leq n-1$, and ...
13
votes
1answer
575 views
A funny factorization of the Jacobian coming from the lines on the Fermat cubic
Here is something which came up in my algebraic geometry class, and I'm wondering if it has a deeper explanation. Let $F(w,x,y,z) = w^3+x^3+y^3+z^3$ and let $X$ be the cubic surface in $\mathbb{P}^3$ ...
2
votes
1answer
103 views
Piercing of subspaces in a projective space?
The "piercing subspace" problem may be stated as follows:
There are given several subspaces in a projective space, rather non-intersecting.
Find an additional subspace of a prescribed dimension that ...
8
votes
0answers
603 views
How many ways can a snake lie?
This is essentially a question about counting nonintersecting short paths in a
cubic lattice, but with a twist. (One constraint that I did not make clear below
is that when to turn is already chosen:...
2
votes
1answer
384 views
degree 7 rational curves through ten points in P4
This is a very classical flavoured question, and probabaly it is not difficult. I would like to know the shape of the space of rational degree 7 curves in $P^4$ that pass through 10 fixed points. By "...
1
vote
1answer
195 views
Curve of degree $d$ through $2d+1$ points in $\mathbb P^3$
It is known that a Hilbert scheme of degree $d$ curves in $\mathbb P^3$ can have dimension more than $4d$. But, does it imply that for some types of curves there are such a curve through any, say, $2d+...
3
votes
1answer
392 views
Counting curves of degree 4 in $\mathbb{P}^{3}$
Let $p_1,...,p_8\in\mathbb{P}^{3}$ be points in linear general position. Then there exists a unique elliptic curve $C$ of degree $4$ passing through $p_1,...,p_8$. I am interested in what happens for ...
9
votes
0answers
691 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....
3
votes
0answers
267 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
1answer
323 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) ...
asked Oct 31 '13 at 20:10
Mohammad Farajzadeh-Tehrani
6,3281 gold badge22 silver badges62 bronze badges
6
votes
1answer
577 views
Nagata's conjecture in positive characteristic
For a $\mathbb C P^2$ is known a result: if through the generic points $p_1,p_2,\dots p_n$ with multiplicities $m_1,m_2\dots, m_n$ correspondingly a degree $d$ irreducible reduced curve passes then $d^...
6
votes
0answers
230 views
Counting plane curves over various fields
Fix two integers $d$ and $g$. The number of genus $g$ and degree $d$ curves passing through $3d+g-1$ generic points on the complex projective plane is finite and doesn't depend on the choice of points....
3
votes
0answers
237 views
Chow ring of a $\mu_2$-gerbe
Suppose that $X$ is a stack, and $Y \to X$ is a $\mu_2$-gerbe. Is there any relationship between the integral Chow rings (in the sense of Edidin and Graham) of $X$ and $Y$?
(I assume they become ...
3
votes
1answer
206 views
“Degree” of a Fano Scheme of a projective variety
Consider subschemes $F$ of the Grassmannian $\mathbb{G}(k,n)$ satisfying the condition that each point of $\mathbb{P}^n$ is contained in only finitely many of the $k$-planes in $F$. Does this give us ...
