Questions tagged [enumerative-geometry]
The enumerative-geometry tag has no usage guidance.
33 questions with no upvoted or accepted answers
11
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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, ...
- 271
9
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0
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860
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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
8
votes
0
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642
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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:...
- 2,158
7
votes
0
answers
157
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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 ...
- 91
6
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237
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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....
- 143
5
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answers
163
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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 ...
- 719
4
votes
0
answers
202
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Cohomological methods in intersection theory and derived categories
Are there any enumerative questions akin to: “What is the number of planes containing a given line tangent to a given cubic surface in $\mathbb{P}^3$” that we can answer using derived categories? I've ...
- 511
4
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212
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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
4
votes
0
answers
105
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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,...
- 17.4k
4
votes
0
answers
107
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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 ...
- 13.4k
3
votes
0
answers
165
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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
3
votes
0
answers
118
views
Divide Euclidean space by surfaces
It is well known that $n$ hyperplanes in $\mathbb{R}^k$ can divide $\mathbb{R}^k$ into at most $p$ regions where $p$ is
\begin{equation}
1 + n + C^2_n + \cdots + C^k_n
\end{equation}
Is there similar ...
- 185
3
votes
0
answers
90
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 ...
- 3,337
3
votes
0
answers
794
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 ...
- 528
3
votes
0
answers
270
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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 ...
3
votes
0
answers
298
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 ...
- 1,832
2
votes
0
answers
143
views
Enumerative geometry and restricted plane partitions
Donaldson-Thomas theory is an enumerative theory for virtual counts of ideal sheaves (with trivial determinant) of the structural sheaf $\mathcal{O}_{X}$ of some smooth projective manifold $X$.
There ...
- 502
2
votes
0
answers
179
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 ...
user114666
2
votes
0
answers
258
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 ...
- 7,736
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
2
votes
0
answers
156
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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 $d$...
- 91
2
votes
0
answers
200
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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 ...
- 443
2
votes
0
answers
154
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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 ...
- 173
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
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
2
votes
0
answers
396
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Noether-Lefschetz locus in enumerative geometry.
It is well known that if you have a smooth quartic surface $X\subset \mathbb{P}^3$, it may or may not have lines in it. Indeed, $X$ has the following options, 64 (the maximal number), 32, 16, or none.
...
- 2,132
1
vote
0
answers
127
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Schubert calculus and the representation ring of the general linear algebra
Schubert calculus studies the structural constants of the standard basis of the cohomology ring of the quantum Grassmannians. It is well known that it is isomorphic to the fusion ring of the category ...
1
vote
0
answers
67
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Counting pieces when an object is cut n ways
I was reading a passage from an old essay by Martin Gardner on the calculus of finite differences, and it seems to me that there is more that can and should be said about seemingly anomalous values of ...
- 19.7k
1
vote
0
answers
292
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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 ...
user35360
1
vote
0
answers
29
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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 $\lambda =[d]$ ...
- 685
1
vote
0
answers
240
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
1
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0
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266
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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
0
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0
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107
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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 ...
- 778