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7
votes
0answers
467 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 ...
2
votes
1answer
297 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 ...
0
votes
1answer
147 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, ...
1
vote
1answer
170 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 ...
6
votes
0answers
164 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 ...
3
votes
0answers
238 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 ...
4
votes
1answer
208 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
1answer
409 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 ...
1
vote
0answers
123 views

Ellipses touching a circle [closed]

This is a crosspost from math.stackexchange, where the question did not find an answer. Given a circle and two points $A$, $B$ in the plane, how do I find an ellipse with focal points $A$ and $B$ ...
5
votes
0answers
164 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 ...
3
votes
0answers
147 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 ...
3
votes
1answer
146 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 ...
3
votes
2answers
387 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 ...
1
vote
1answer
232 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 ...
7
votes
1answer
530 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 ...
3
votes
1answer
497 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 ...
2
votes
0answers
143 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
votes
1answer
295 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 ...
8
votes
1answer
419 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). ...
5
votes
2answers
1k views

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
votes
1answer
501 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 ...
0
votes
1answer
244 views

What is the simplest way to show that a section of a vector bundle is transverse to the zero set

Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$, be the space of homogeneous degree $d$ polynomials in three variables $[X,Y,Z] \in \mathbb{P}^2$ upto scaling, where $\delta_d = \frac{d(d+3)}{2}$. ...
5
votes
3answers
524 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 ...
12
votes
3answers
943 views

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. ...
6
votes
2answers
379 views

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 ...
9
votes
1answer
1k views

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 ...
0
votes
1answer
398 views

Does any one understand the details of M Kazarian's work in enumerative geometry of $\mathbb{C}\mathbb{P}^2$ ?

I wanted to know if anyone understood the details of the paper "Multisingularities, cobordisms, and enumerative geometry" available at the site http://www.mi.ras.ru/~kazarian/. In particular does ...
5
votes
2answers
492 views

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 ...
5
votes
2answers
333 views

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 ...
1
vote
1answer
362 views

Computing 3 points Gromov-Witten invariants of the Grassmannian

This is from an exercise in Koch, Vainsencher - An invitation to quamtum cohomology. Background The exercise asks to compute the 3-points Gromov-Witten invariants of the Grassmannian $G = ...
6
votes
1answer
688 views

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 ...
2
votes
0answers
248 views

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. ...
5
votes
4answers
515 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 ...
10
votes
5answers
888 views

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 ...
17
votes
7answers
1k views

Higher Dimensional Gromov-Witten Theories

So, 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" the curve of genus $g$ ...