# 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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### Maximum number of intersection points of two different Bernoulli lemniscates

What is the maximum number of intersection points of two different Bernoulli lemniscates in the real plane? (Of course two identical lemniscates share an infinite number of points.) Here are some of ...
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### Can Yoneda lemma for smooth projective varieties only use curves?

Let $X,Y$ be two projective smooth varieties over an algebraically closed field $k$, if we know functorial isomorphism $Hom(C,X) \cong Hom(C,Y)$ as sets for every smooth projective curve $C$, do we ...
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### Do Neron models of hyperbolic curves exist

Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Assume that $g\geq 2$. Does there exist a Neron model $\mathcal X$ for $X$ over $O_K$? By a Neron model, I mean ...
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### How to write holomorphic line bundles on compact Riemann surfaces in coordinates

I would like to write down coordinate charts for holomorphic vector bundles on a smooth complex (compact) curve $C$. For example, if $C = \mathbb C \mathbb P^1 = \{ [x_0 : x_1 ] \}$, let \begin{...
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### Given a curve over $\mathbb{P}^1$ how can I determine the monodromy around a ramification point?

Given a smooth curve of the form $$f(y) - g(x)$$ over $\mathbb{A}^1_x$ with a smooth compactification $C \to \mathbb{P}^1$, how can I determine the monodromy of the points of the fibers? Are there ...
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### Relationship between twisting sheaves and divisor sheaves

I'm not really entirely sure how to think about Serre's twisting sheaves $\mathscr{O}(i)$ - on any $\text{Proj}$ construction, really, but let's stick to something like $\mathbb{P}_2$ for now for ...
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### Why do algebraic varieties contain curves passing through two given points

Let $X$ be an algebraic variety over the complex numbers. My definition of an algebraic variety is a finite type separated $\mathbf C$-scheme. Someone told me that such varieties have the following ...
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### A question on Newton's “theorem about ovals”

This is a question about a result from Newton's Principia. It says, roughly, that the if you intersect lines $ax + by + c$ with a smooth, closed, convex curve, then the area of the curve that the line ...
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### Error in Algebraic Curves by Fulton?

The following lemma is from section 3.3 of Fulton's algebraic curves. Notation: $\psi$ is a map from $k[X,Y]/I^n×k[X,Y]/I^m$ to $k[X,Y]/I^{m+n}$ I'm having some difficulty understanding the ...
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### Affine plane curves classification

Define an affine plane conic as $\operatorname{Spec}A$ where $A=k[x,y]/(f)$ and $f$ a quadratic polynomial with no multiple factors. Define an equivalence relation on the set of alline plane conics ...
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### Katz-Mazur chapter 1 AG questions

I was reading through Katz-Mazur "Arithmetic Moduli of Elliptic Curves", Chapter 1, and ran into some small issues (which might have a lot to do actually with notation). I think most of them are due ...
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### What is an example of a genus 5 smooth projective curve?

Recall that if $k$ is an algebraically closed field, then any degree $d$ plane curve $X$ will have arithmetic genus $g=(d-1)(d-2)/2$, by a simple calculation of $H^1(X,\mathscr{O}_X)$. This tells us ...
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