# Tag Info

Accepted

### Normalization of nodal singularity of a curve and canonical divisor.

I will assume your nodal curve is embedded in $\mathbb{P}^2$. Then blowing up $\mathbb{P}^2$ at the nodal point will exactly give the normalization of our curve (by strict transformation). Let me call ...
• 656
Accepted

### Smoothing projective nodal curve, is the general fiber smooth?

Let $f:X \to T$ be a flat and proper morphism, locally of finite presentation. By Tag 01V8, $f$ is smooth if and only if the fibers of $f$ are smooth. The set of points in $X$ where $f$ is not smooth ...
• 5,235
Accepted

### What is a curve at $y=\infty$ mean?

It is explained to some extent in the link. Work in the projective plane, so introduce a variable $z$ and make every term in the polynomial homogeneous of degree 3: F(x,y,z) = xyz + ax^3 + bx^2z + ...
• 117k
1 vote

1 vote

### How to show there are only two discrete valuation rings with quotient field $k(x)$?

The case where $k[x] \subseteq S$ is easy. Let $m$ be maximal ideal of $S$. Thus $k[x] \cap m$ is a prime ideal in $k[x]$. Since $k[x]$ is a PID, $k[x] \cap m$ is either zero ideal or a maximal ideal. ...

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