# 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.

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### For every function field $L$ there is a smooth projective curve $C$ with $L\simeq k(C)$

Let $k$ be an algebraically closed field. It is a well-known result that: The category of smooth (i.e., non-singular) projective curves with dominant morphisms is equivalent to the category of ...
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### Proof of $Z(I(X))=X$ for $X\subset \mathbb{P}^n$ a projective algebraic set.

Let $X\subset \mathbb{P}^n$ be a projective algebraic set. That $X \subset Z(I(X))$ is clear. For the other inclusion, let's take $x\in Z(I(X))$ and we want to see that $x\in X$. Since $x\in Z(I(X))$,...
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### Smooth special fiber implies smooth generic fiber when proper and flat

Let $f : X \rightarrow S$ be a morphism between schemes, with $f$ locally of finite presentation, proper and flat, and $S$ the spectrum of a discrete valuation ring with closed point $s$. Assume that ...
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### Is it true that $P_f(k)\simeq C_{~^df}(k)\sqcup C_{~^d f_p}(k)$?

Let $f\in k[x_1, \ldots, x_{n+1}]$ be an homogeneous degree $p$ polynomial and let $~^d\! f\in k[x_1, \ldots, x_n]$ be defined by: $$~^d\! f(x_1, \ldots, x_n):=f(x_1, \ldots, x_n, 1).$$ Let $P_f(k)$ ...
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### Topological Euler characteristic of a cycle of algebraic curves

The topological Euler characteristic of a smooth algebraic curve of genus $g$ equals $2-2g$. Let $C=\bigcup_{i=1}^n C_i$ be a union of smooth algebraic curves. I am wondering if there is a way to ...
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### Characterize the values of A and B for which variety is singular

Let $A, B ∈ \overline{K}$ with $K$ a field. Characterize the values of A and B for which each of the following variety is singular: $V : Y^2 Z + AXY Z + BY Z^2 = X^3$ Looking at the point where ...
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### Notation for Divisors in Liu's AG & AC

I have a question about a notation in Liu's "Algebraic Geometry and Arithmetic Curves" in the excerpt below (or look up at page 276): What is here means by $(f)_0$ and $(f)_{\infty}$? Does anybody ...
For a given curve: $$C: \frac {ax^2+bx+c}{dx+e}$$ where $a,b,c,d,e$ are integers. Let $f(x)=ax^2+bx+c$ . Oblique asymptote can be found by long division of numerator by denominator.Here oblique ...
Since an elliptic curve given by the homogenized polynomial $$y^2z=x^3+axz^2+bz^3$$ is a plane projective curve, we can get its dual. From this Wikipedia link, eliminating $p$, $q$, $r$, and $λ$ from ...