# Questions tagged [algebraic-curves]

An algebraic curve is an algebraic variety of dimension one. An affine algebraic curve can be described as the zero-locus of $n-1$ independent polynomials of $n$ variables in affine $n$-space over a field. Examples include conic sections, compact Riemann surfaces and elliptic curves. Singularities of curves are extensively studied as a basic case in singularity theory. Via algebraic function fields and modular curves they have links to number theory.

2,520 questions
Filter by
Sorted by
Tagged with
4k views

### Geometric interpretation of the Riemann-Roch for curves

Let $X$ be a smooth projective curve of genus $g\geq2$ over an algebraically closed field $k$ and denote by $K$ a canonical divisor. I have some clues about the geometrical interpretation of the ...
• 6,947
7k views

### Intuitive explanations for the concepts of divisor and genus

When trying to explain AG-codes to computer scientists, the major points of contention I am faced with are the concepts of divisors, Riemann-Roch space and the genus of a function field. Are there any ...
• 778
4k views

• 3,676
2k views

### How to compute the order $\text{ord}_P (f)$ for $f \in K(C)$

First lets fix some notation. Let $C$ be a projective curve (i.e. projective variety of dimension 1) defined over a field $K$. Suppose that $P \in C$ and that $P$ is a smooth point. It is known that ...
• 15.2k
896 views

### Why do varieties with torsion canonical sheaf have finite etale covers with trivial canonical sheaf

Let $B$ be a variety with torsion canonical sheaf, i.e., $\omega^{\otimes n}_B \cong \mathcal O_B$ for some $n>0$. Then, why does there exist a finite (etale?) morphism $X\to B$ such that $K_X$ is ...
• 647
4k views

### Canonical divisor on algebraic curve

Can someone help me with this problem? Let $D$ be a divisor on an algebraic curve $X$ of genus $g$, such that $\deg D = 2g-2$ and $\dim L(D) = g$. Then $D$ must be a canonical divisor. By Riemann-...
• 6,788
4k views

### What is normal crossing?

I could not find any reference for normal crossings. The definition here is not so clear to me. In some texts, they sometimes said that two varieties have normal-crossing (non-normal crossing) with ...
• 959
865 views

### Precise definition of an "algebraic function"

Remark. I'd like to avoid the "ring of formal expressions" viewpoint for this question. I know we can avoid these kinds of questions by working "purely algebraically" and in particular by taking the ...
• 68k
661 views

• 5,345
304 views

### Finding an example of a degree 5 rational curve in $\mathbb{P}^3$ which is not in a quadric.

This is exercise IV.6.2 in Hartshorne, which asks us to find an example of a nonsingular rational degree $5$ curve in $\mathbb{P}^3$ which is not contained in a quadric. I'm somewhat at a loss for how ...
• 5,554