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I am reading Pandharipande and Thomas's article:curve counting via stable pairs in the derived categories. In their page 22, they gives following statement: Statement: let $I_n(X,\beta)$ be the ...
Lin Sen's user avatar
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This question is basically to get some motivation and a birds eye view of the subject Donaldson-Thomas(DT) theory in algebraic geometry. I was looking some references in the subject. It seems most of ...
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The conjecture by Kenyon-Okounkov-Sheffield says that given uniformly chosen lozenge tiling of a hexagon: you can express the random height function of the hexagonal graph (viewing the above picture ...
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I was this past year working with a bright high-schooler on algebraic geometry following Reid's book Undergraduate Algebraic Geometry, and we got all the way to proving that there is at least one line ...
David Roberts's user avatar
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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&...
user302934's user avatar
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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 $\mathcal{E} = \{E_i\}_{i\in I}$...
Reginald Anderson's user avatar
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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 $n-2$...
Simon L Rydin Myerson's user avatar
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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 ...
Hao Yu's user avatar
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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 ...
user290289's user avatar
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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 ...
Reginald Anderson's user avatar
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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 ...
Puzzled's user avatar
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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 ...
Didier de Montblazon's user avatar
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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 ...
James Propp's user avatar
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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 ...
Svjetlan Feretic's user avatar
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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 ...
Mohammad Farajzadeh-Tehrani's user avatar
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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 ...
Ramiro Hum-Sah's user avatar
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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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$\DeclareMathOperator\Gr{Gr}$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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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 ...
AG practitioner's user avatar
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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 ...
Anon's user avatar
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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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Akira Toriyama
4 votes
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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 ...
user avatar
2 votes
1 answer
713 views

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-...
Ritwik's user avatar
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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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330 views

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 ...
Asvin's user avatar
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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 ...
Y. Zhao's user avatar
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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 $\lambda =[d]$ ...
GGT's user avatar
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254 views

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 ...
Hans Sachs's user avatar
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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 ...
GGT's user avatar
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10 votes
1 answer
613 views

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 ...
Pierre Dubois's user avatar
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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/...
Ritwik's user avatar
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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,...
Matthias Wendt's user avatar
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6 votes
2 answers
611 views

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-...
Rene Schipperus's user avatar
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1 vote
1 answer
340 views

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 ...
Q. Q.'s user avatar
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3 votes
2 answers
301 views

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 ...
john mangual's user avatar
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23 votes
4 answers
3k views

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 ...
Saal Hardali's user avatar
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7 votes
1 answer
411 views

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?
forget this's user avatar
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2 votes
1 answer
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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 &...
heinrich's user avatar
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11 votes
0 answers
412 views

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, ...
Complexification's user avatar
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1 vote
0 answers
290 views

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. $$ ...
Nati's user avatar
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3 votes
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953 views

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 ...
A. S.'s user avatar
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4 votes
0 answers
135 views

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 ...
Wolfgang's user avatar
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2 votes
0 answers
174 views

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 $d$...
peter's user avatar
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2 votes
0 answers
251 views

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 ...
Han Jin Ma's user avatar
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2 votes
0 answers
180 views

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 ...
User133713's user avatar
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12 votes
1 answer
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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 ...
Ritwik's user avatar
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1 vote
0 answers
291 views

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 ...
Ritwik's user avatar
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7 votes
1 answer
823 views

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 $$ \beta := n L + m_1 E_1 + \ldots + m_k E_k $$ ...
Ritwik's user avatar
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4 votes
1 answer
384 views

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 $$ \beta := n L + m_1 E_1 + \ldots + m_k E_k $$ ...
Ritwik's user avatar
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