Questions tagged [algebraic-curves]

Question 1

Let $C$ be a nice (i.e. smooth, projective, and geometrically integral) curve of genus $g \geq 3$, defined over some field $k$. I am mainly interested in the case that $k$ is a number field, but feel ...

Question 2

Cubic Curve to Weierstrass Form For the cubic curve $C$ in general form with rational coefficients:$$ax^3+bx^2y+cxy^2+dy^3+ex^2+fxy+gy^2+hx+ky+l=0,$$we are interested in finding rational points on it. ...

Question 3

Suppose $C$ is the plane curve cut out by the equation $$ u^2 + \frac{8}{3} uv + v^2 = \frac{5}{3} - u^2 v^2 . $$ Can you provide an explicit change of coordinates that puts this curve in Edwards form,...

Question 4

Let $S$ be a smooth complex projective surface. We consider the double cover $\pi:X\to S$ branched along $B\equiv 2L$(suppose $B$ is smooth). Now let $C\subset X$ be an irreducible curve. If $C$ is ...

Question 5

I consider the Hermitian function field $H = \mathbb{F}_4(x,y)$, given by $y^2 + y = x^3$, which is a quadratic extension of $F = \mathbb{F}_4(x)$. Let $P$ be a place of $F$. If $v_P(x^3) \ge 0$, then ...

Question 6

Let $X$ be a compact Riemann surface and suppose the zero mean theorem holds: Zero mean Theorem (p.318 Miranda):If $X$ is an algebraic curve and $\eta$ is a $C^\infty(X)$ $2-$form on it, then there ...

Question 7

Let $L$ be a lattice in $\Bbb{C}$, and let $\pi$ be the natural protection. Show that $dz,d\bar{z}$ are well-defined holomorphic $1-$forms on $X=\Bbb{C}/L$. Attempt: I used charts because 2 are enough ...

Question 8

This is a problem I found on the Rick Miranda's book. Problem What is the minimum integer $k$ such that for every curve $X$ of a fixed genus $g$ there is a holomorphic map $F: X \rightarrow \mathbb{P}^...

Question 9

I'm taking a course in Algebraic Curves and I've come across the Zariski topology, a topology which the set of closed sets are the algebraic sets of $\mathbb{K}^n$. I want to prove that it does indeed ...

Question 10

This is a question my friend raised and we have had difficulty solving it. Suppose that the graph of the polynomial function $f(x)=x^4$ is drawn on a plane. Can we construct the $y$-axis of this ...

Question 11

I am working with parametrized polynomial plane curves and I have several related questions about Milnor numbers and singularities. I would be grateful for references, precise statements (theorem ...

Question 12

This may be a bit vague, but how does one describe vector bundles on nodal cubic such as $C:y^2-x^2-x^3 = 0$? If $E$ is a rank $r$ holomorphic vector bundle on $C$ then given the normalization $\nu : \...

Question 13

I was reading this interesting article from arXiv Relationships Between Hyperelliptic Functions of Genus 2 and Elliptic Functions. On page 9 and 10 it talks about the morphisms between curves of genus ...

Question 14

The isogeny class (over a field $K$) of an elliptic curve $E$ defined over $K$ is the set of all isomorphism classes of elliptic curves defined over $K$ that are isogenous to $E$ over $K$. This ...

Question 15

I am asking whether the following statement is true: given a prime $p$ a finite group $G$ of order coprime to $p$ an integer $g$ greater than 1 an algebraically closed field $K$ of characteristic $p$...

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Questions tagged [algebraic-curves]

When does a non-hyperelliptic curve allow a hyperelliptic cover?

Down-to-earth definition of "genus".

Converting an elliptic curve to Edwards form

Preimage of curves in a double cover of surface

Do places ever become inert in the Hermitian function field over $\mathbb{F}_4$ ($q = 2$)?

Problem out of Miranda on Zero mean theorem

IV.I.$1$ out of Miranda Algebraic curves & Riemann Surfaces.

Minimum degree of a holomorphic map from an algebraic curve of genus g to the Riemann sphere

Proof that the Zariski topology is indeed a topology?

Given the graph $y=x^4$, can we construct the $y$-axis using only a straightedge and a compass?

Milnor number parity and singularities of parametrized plane curves

Vector bundles on nodal curves

Problems with morphisms of algebraic curves from an article

Is an isogeny class of a complex elliptic curve a finite set?

A question about coverings of the projective line in positive characteristic

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