Questions tagged [enumerative-geometry]
The enumerative-geometry tag has no usage guidance.
85 questions
23
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7
answers
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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
23
votes
4
answers
3k
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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 ...
- 7,789
17
votes
1
answer
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Incidence Correspondence
A useful tool in Algebraic Geometry is the incidence correspondence. Loosely speaking, it is a set of the form $$\{(p,X): p \text{ a fixed dimension subscheme of } Y \text{ and } X \text{ a specific ...
- 565
13
votes
1
answer
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Who streamlined Kontsevich's count of rational curves?
Let $N_d$ denote the number of rational curves in $\mathbf P^2$ passing through $3d-1$ points in general position. Maxim Kontsevich discovered a famous recursion for these numbers:
$$ N_d = \sum_{k+l =...
- 40.2k
13
votes
1
answer
605
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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 $\mathbb{P}^3$ ...
- 156k
12
votes
3
answers
1k
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Counting restricted polyominoes
I would like to count the polyominoes of $n$ squares that satisfy several restrictions:
Each is convex: every horizontal, or vertical line, meets the shape in either a single segment,
or not at all.
...
11
votes
2
answers
1k
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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
11
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2
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557
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Hypersurface of singular plane cubics
In the projective space $\mathbb{P}^9 = \mathbb{P}(\mathbb{C}[x,y,z]_3)$, parametrizing plane cubics, consider the hypersurface $X\subset\mathbb{P}^9$ whose points corresponds to singular cubics. The ...
- 8,998
11
votes
5
answers
2k
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Classical Enumerative Geometry References
I want to start out by making this clear: I'm NOT looking for the modern proofs and rigorous statements of things.
What I am looking for are references for classical enumerative geometry, back before ...
- 16k
11
votes
1
answer
706
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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
11
votes
0
answers
358
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, ...
- 271
10
votes
1
answer
559
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 ...
- 643
10
votes
1
answer
891
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 ...
- 1,803
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
8
votes
1
answer
566
views
Multiple Hodge integrals and integrability
It is known that a generating function of the linear Hodge integrals is a tau function of the KP hierarchy, namely a one-parameter deformation of the Kontsevich-Witten tau-function (see Kazarian). ...
- 1,343
8
votes
0
answers
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
1
answer
293
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?
- 615
7
votes
1
answer
705
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
7
votes
0
answers
157
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 ...
- 91
6
votes
4
answers
869
views
Interaction of topology and the Picard group of Algebraic surfaces
It is well known that a smooth cubic surface $X\subset \mathbb{P}^3$ has exactly 27 lines in it. Furthermore, it is easy to check that Picard group $$Pic(X)\cong \mathbb{Z}^7$$ Here the generators are ...
- 2,132
6
votes
2
answers
1k
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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?
6
votes
2
answers
479
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How do the number of plane curves over a finite field of a fixed genus increase with the degree?
Let $k$ be a finite field. We define $N(d,g)$ to be the number of plane curves $f(x,y)$ defined over $k$ of degree $d$ with (geometric) genus $g$. If $D(d) := (d-1)(d-2)/2$ (the maximum possible ...
- 4,636
6
votes
2
answers
451
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Up to projectivities, which configurations of four lines in $\mathbb{P}^3$ can one distinguish?
Background
I am interested in the projective classification of reduced curves of degree four in $\mathbb{P}^3(\mathbb{R})$ (and more generally of degree $n+1$ in $\mathbb{P}^n(\mathbb{R})$). More ...
- 95
6
votes
2
answers
506
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-...
- 953
6
votes
1
answer
599
views
Embedding $G(2,n)$ into $G(k,n)$
Let
$$M=\begin{pmatrix}
u_1 & u_2 & \ldots & u_n \\
v_1 & v_2 & \ldots & v_n \\
\end{pmatrix}$$
be a $2 \times n$ matrix. Define $\nu(M)$ to be the $k \times n$ matrix
$$\nu(M)...
- 156k
6
votes
1
answer
693
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^...
- 5,043
6
votes
1
answer
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Chow Ring of Moduli Space of Abelian Varieties
Is there a good reference for the structure of the Chow ring of $\mathcal{A}_g$, the moduli space of complex principally polarized abelian varieties? More generally, references for the intersection ...
- 16k
6
votes
0
answers
237
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....
- 143
5
votes
2
answers
647
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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
5
votes
3
answers
597
views
Thom polynomial for contact algebraic structures
Let's consider a algebraic contact structure $P$ on $\mathbb CP^3$
and a algebraic curve $C$ degree $d$ and genus $g$. Let's assume
that contact structure has degree $p$ (see
Polynomial contact ...
- 5,043
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) ...
5
votes
2
answers
2k
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Nagata's conjecture, Seshadri constant
What is it known now about Nagata's conjecture and Seshadri constant (http://en.wikipedia.org/wiki/Nagata%27s_conjecture_on_curves and http://en.wikipedia.org/wiki/Seshadri_constant) for toric ...
- 5,043
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
5
votes
0
answers
163
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 ...
- 719
4
votes
2
answers
583
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
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 $$
...
- 3,245
4
votes
1
answer
384
views
Hurwitz numbers and Frobenius manifolds
Generating functions of the Gromov-Witten invariants (as well as some other important partition functions) are known to be related to the Frobenius manifold structure. Are there any Frobenius ...
- 1,343
4
votes
0
answers
202
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Cohomological methods in intersection theory and derived categories
Are there any enumerative questions akin to: “What is the number of planes containing a given line tangent to a given cubic surface in $\mathbb{P}^3$” that we can answer using derived categories? I've ...
- 511
4
votes
0
answers
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How does one obtain the formula for the number of genus one curves in P^2 using Getzler's relation?
I am trying to get the formula for the number of degree $n$ genus one curves in $\mathbb{P}^2$ passing through 3n generic points, by following the procedure in Getzler's paper
https://arxiv.org/pdf/...
- 3,245
4
votes
0
answers
105
views
Closed form for integer series from enumerative geometry problem?
Is there a closed form for the following integer sequence:
$$
1,6,145,8806,830622,100317140,14342519633,2325250316950,...
$$
This is the degree of the $2n$-th power of the Schubert class $\sigma_{2,...
- 17.4k
4
votes
0
answers
107
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 ...
- 13.4k
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
3
votes
1
answer
707
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 ...
- 8,998
3
votes
2
answers
743
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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
2
answers
238
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 ...
- 22.8k
3
votes
1
answer
219
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 ...
- 265
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
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