# Tagged Questions

**3**

votes

**1**answer

214 views

### Conditions for a parametric curve to avoid self-intersection?

Suppose a planar curve $C$ is defined by parametric
equations in $t$: $x(t)$ and $y(t)$.
Are there conditions on these two functions that guarantee
that $C$ does not self-intersect?
For example,
the ...

**12**

votes

**2**answers

702 views

### Permuting collinear points on a curve

Let $C \subset {\bf CP}^2$ be an irreducible algebraic smooth (projectively) planar curve over the complex numbers of degree $d$ (we allow finitely many points to be deleted from $C$ to make it ...

**2**

votes

**1**answer

224 views

### Calculating the local index of intersection of two algebraic curves.

Let $F_1,F_2$ be two polynomials of two variables $(x,y)$ (say complex variables).
Suppose that $F_1$ and $F_2$ have no common factors and $F_1(P)=F_2(P)=0$.
What is in practice the quickest way to ...

**16**

votes

**1**answer

545 views

### Do mapping classes have gonality?

(This question was discussed by people at the PCMI workshop on moduli spaces, without any clear resolution, so I thought I'd throw it open to MO.)
The hyperelliptic mapping class group is (by ...

**11**

votes

**4**answers

1k views

### What is the Euler characteristic of a Hilbert scheme of points of a singular algebraic curve?

Let $X$ be a smooth surface of genus $g$ and $S^nX$ its n-symmetrical product (that is, the quotient of $X \times ... \times X$ by the symmetric group $S_n$). There is a well known, cool formula ...

**6**

votes

**1**answer

291 views

### Reference for equivalent definitions of the genus

Let $X$ be a (edit: nonsingular) projective complex algebraic curve. The genus of $X$ can be defined as the dimension of the space of holomorphic $1$-forms on $X$, which in turn can be defined either ...

**25**

votes

**3**answers

2k views

### Mumford conjecture: Heuristic reasons? Generalizations? … Algebraic geometry approaches?

The Mumford conjecture states that for each integer $n$, we have: the map $\mathbb{Q}[x_1,x_2,\dots] \to H^\ast(M_g ; \mathbb{Q})$ sending $x_i$ to the kappa class $\kappa_i$, is an isomorphism in ...

**8**

votes

**5**answers

2k views

### Ribbon graph decomposition of the moduli space of curves

What is a ribbon graph? What is the ribbon graph decomposition of the moduli space of curves? What are some good references for this material?