# Questions tagged [enumerative-geometry]

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### Examples of nontrivial configurations of rational curves of degree $\leq 3$ in the projective plane

Consider the complex projective plane $P^2$. A rational curve in $P^2$ of degree $\leq 3$ is either a line, a smooth conic, a nodal cubic, or a cuspidal cubic. I am looking for some "nontrivial&...
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1 vote
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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 $\mathcal{E} = \{E_i\}_{i\in I}$...
1 vote
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### Degree three, codimension one subvarieties lying on a quadratic hypersurface

Let $H$ be an irreducible hypersurface in $\mathbb P^n$ of large-ish degree, say 14. This question is about subvarieties $V$ of $H$ such that $V$ has codimension 1 in $H$ (i.e. $V$ has dimension $n-2$...
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### 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 $$1 + n + C^2_n + \cdots + C^k_n$$ Is there similar ...
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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 ...
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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 ...
536 views

### 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 ...
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1 vote
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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
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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 ...
• 19.6k
1 vote
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### Rotational invariance assumed, what is the number of $r$-sided simple polygons that can be inscribed into an $n$-sided regular polygon?

When I say that an $r$-sided simple (i.e., not self-intersecting) polygon is inscribed into an $n$-sided regular polygon, I mean that every vertex of the simple $r$-gon is also a vertex of the regular ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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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 ...
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### 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 ...
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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 ...
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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?
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### 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 ...
598 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-...
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1 vote
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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 ...
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### 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 ...
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### 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 ...
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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 $\lambda =[d]$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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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/...
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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, ...
1 vote
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### 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 ...
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 ...
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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 ...
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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 $\mathbb{P}^3$ ...