# Questions tagged [enumerative-geometry]

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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?
237 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 ...
284 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-...
109 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 ...
140 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 ...
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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 ...
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 $\lambda =[d]$ ...
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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 ...
68 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 ...
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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 ...
161 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/...
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### 3264 rational conics

It's a classical fact that there are 3264 plane conics tangent to 5 general conics, over $\mathbb{C}$ . It was also shown that the 3264 can be defined over the reals  or . More precisely, ...
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. $$... 0answers 359 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 ... 0answers 92 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 ... 0answers 140 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... 0answers 182 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 ... 0answers 115 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 ... 1answer 565 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 ... 0answers 245 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 ... 1answer 522 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 $$... 1answer 273 views ### What are the indecomposable classes on a del-Pezzo surface? Let X_k be \mathbb{P}^2 blown up at k points (where k is 0 to 8). Let \beta \in H_2(X_k, \mathbb{Z})  be a homology class given by$$ \beta := n L + m_1 E_1 + \ldots + m_k E_k $$... 0answers 142 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 ... 1answer 217 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 ... 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 ... 1answer 558 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 ... 1answer 102 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 ... 0answers 600 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:... 1answer 376 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 "... 1answer 192 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+... 1answer 335 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 ... 0answers 676 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.... 0answers 265 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 ... 1answer 318 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) ... 1answer 545 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^... 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.... 0answers 225 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 ... 1answer 201 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 ... 2answers 582 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 ... 1answer 263 views ### Riemann-Roch and dim of deformation space. Let's consider curve C\subset \mathbb P^n of degree d and genus g. We want to calculate dimension of deformation space of C, i.e. h^0(C,L) where L is the normal bundle. We can decompose ... 1answer 2k views ### 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 ...
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 ...