# Tagged Questions

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 ...

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### multiple tangent lines to a plane curve

Suppose that $C=V(F)$, with $F\in k[X,Y]$ ($k$ algebraically closed), is an irreducible plane curve such that $P=(0,0)$ is a nodal singular point (or an ordinary multiple point according to Fulton's ...
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### The degree of a map onto a rational normal curve

Source: Miranda's Exercise J Page 167 If $v^{2}=h(u)$ defines a hyperelliptic curve of genus $g$, then $\phi=[1:u:u^{2}:\dots,u^{g-1}]$ defines a degree $2$ map onto a rational normal curve of ...
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### Map of smooth curves and its separability degree

I'm interested in a proof of the following fact from Silverman: Arithmetic of Elliptic Curves: Let $\Phi: C_1 \rightarrow C_2$ be a nonconstant map of smooth curves. Then for all but finitely many ...
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### Computing a quotient of rings

Let $R=k[x,y]/(y^2-x^2-x^3)$ and $I=(x,y)\cdot R \subset R$. I would like to show that $$\bigoplus_{i=0}^{\infty} I^i\,/\,I^{i+1} \cong \,k[x,y]\,/\,(x^2-y^2).$$ Could you please help me? Remark: ...
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### About Linear Systems on Curves.

Let $C$ be a smooth irreducible (complex) curve of genus $g\geq2$. The gonality of $C$ is defined as the minimum degree of surjective morphisms $C\rightarrow\Bbb{P}^1$. So $C$ has gonality $d$ if it ...
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### Is this union of tangent spaces a known object in Algebraic Geometry?

Let $C$ be a family of smooth (all but one curve is smooth, I'll explain later) cubic curves in $\mathbb{P}_{\mathbb{C}}^2$ with the following property: given any two elements $a,b \in C$, the curves ...
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### Dimension of a meromorphic differentials space

What is the dimension $d_{ \large k,n}$ of the space of the degree $k$ meromorphic differentials on the sphere with fixed residues ($\alpha_i$) at $n$ points $z_i$ ? The question is asked in this ...
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### Tate's Thesis: in what sense is Tate's Theorem 4.2.1 the Riemann-Roch theorem for curves?

I am reading Tate's Thesis. Tate derives a theorem which he calls "the number-theoretic analogue of the Riemann-Roch theorem" from an abstract Poisson summation formula. I am accustomed to thinking of ...
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### Compact Riemann surfaces are projective varieties.

We know that every compact Riemann surface is a complex compact manifold of dimention one. But why every compact Riemann surface is a projective variety?
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### $V(X^m + Y^m - Z^m)$ (projective Fermat curve) isomorphic to projective line iff $m=1, 2$

I've convinced myself that the projective Fermat curve $V(X^m + Y^m - Z^m) \subset \mathbb{P}^2$ is isomorphic to a projective line if and only if $m =1$ or $m = 2$, but I'm not sure how to prove this ...
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### Linear system of degree $d$ curves passing $m$ times through $P$ in the blow-up at $P$.

Given a point $P$ in $\mathbb{P}^2$ and a natural number $m$ we consider the linear system $\mathcal{L}$ of curves of degree $d$ passing $m$ times through $P$. If $H$ is the line class of the plane, ...
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### Dimension of family of hyperelliptic curves

Suppose we have an elliptic curve E with a point $P$ of order $5$ over a field of characteristic $0$. Denote $E'$ the curve $E/\langle P\rangle$. Now let $x$ (resp. $x'$) be a function on $E$ (resp. ...
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### If $f(u(x), v(y))=f(x, y)$, can we conclude that either $u(x)=x$ or $v(y)=y$?

Suppose $k$ is an algebraically closed field, and $f\in k[x, y]$ is an irreducible polynomial in two variables. Furthermore, suppose that $f(u(x), v(y))=f(x, y)$ for every $x, y\in k$, where ...
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### How prove this for sufficiently large $n$,the functions $x(t)^{i}y(t)^{j},0\le i,j\le n$ are linearly dependent

An algebraic curve in $R^2$ is the locus of zero of a polynomial $f(x,y)$ in two variables By a polynomial path in $R^2$,we mean a parametrized path $x=x(t),y=y(t)$,where $x(t),y(t)$ are ...
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### kernel of the norm map of jacobians

Given an étale double covering of curves $f: C\to C_0$, there is an induced norm map $Nm: J(C) \to J(C_0)$, which sends $\sum_i p_i$ to $\sum_i f(p_i)$. On page 285 of the book Geometry of algebraic ...
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### Maps between skyscraper sheafs and line bundles

Let $C$ be a smooth projective curve over $\mathbb{C}$, $x \in C$ is a point, $k(x)$ is a skyscraper sheaf at his point, $L$ is some line bundle, why $$Hom(k(x), L) \cong 0 ?$$ Is this (or some ...
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### Explicit Derivation of Weierstrass Normal Form for Cubic Curve

In page 22-23 of Rational Points on Elliptic Curves by Silverman and Tate, authors explain why is it possible to put every cubic curve into Weierstrass Normal Form. Here are relevant pages: (My ...
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### Rational functions on $V(xw-yz)$

Problem: Let $k$ be an algebraically closed field and $V=V(xw-yz)=\{(x,y,z,w)\in\mathbb{A}^4(k): xw-yz=0\}$. Let $\Gamma(V)$ be the ring of coordinates of $V$ and $k(V)$ its field of fractions. Let ...
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### Equations for double etale covers of the hyperelliptic curve $y^2 = x^5+1$

Let $X$ be the (smooth projective model) of the hyperelliptic curve $y^2=x^5+1$ over $\mathbf C$. Can we "easily" write down equations for all double unramified covers of $X$? Topologically, these ...
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### Comparing 2 non-linear curves

I have 2 non-linear curves having (x,y)values. The x values are varying from 0 to 127 in both the curves and y values are of different magnitude for 2 curves. How can I compare these 2 non-linear ...
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### Fundamental group of a complex algebraic curve residually finite?

Is the analytic fundamental group of a smooth complex algebraic curve (considered as a Riemann surface) residually finite?
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### automorphism groups of hyerelliptic curves in positive charactersitic

It appears that the automorphism groups of hyperelliptic curves are at least well studied, if not understood, in the characteristic zero case. I would imagine that most of these results would carry to ...
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### Group actions on Čech cohomology

Suppose we have a curve $X$ and a group $G$ acting on $X$. Then one has an induced action of $G$ on the sheaf cohomology of $\mathcal O_X$. I wondered what one can say about the group action on the ...
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### Genus of a function field

There is a one-to-one correspondence between isomorphism classes of smooth absolutely irreducible curves $X/\mathbb{k}$ and isomorphism classes of ﬁelds $\mathbb{K}$ of transcendence degree $1$ over ...
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### Computing algebraic de-Rham cohomology via Čech cohomology

I have been reading this paper about de-Rham cohomology of hyperelliptic curves, and I have been trying to recompute some of what has been done in section 3. In particular, I am trying to see why ...
Let $G$ be a finite group, not necessarily abelian. Is there any smooth algebraic curve $C$, with an action of $G$ on $C$, such that the natural quotient map $C \to C/G$ is branched at precisely one ...
Let $k$ be an algebraically closed field, $C/k$ and $C'/k$ be smooth projective curves, and $C'/k \rightarrow C/k$ be a $k$-morphism which is corresponding to the field extension \$k(C) \hookrightarrow ...