# 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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### A question on partial-derivative

Let $f(x,y) = \sqrt {{x^2} + {y^2}} p(x,y) + q(x,y)$, and $p$ and $q$ are two polynomials(non zero) Is it true that; $\frac{{\partial f}}{{\partial x}}$ and $\frac{{\partial f}}{{\partial y}}$ don't ...
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### How can we prove, by Bézout's theorem, that $L$ has finitely many singularities?

Bézout's theorem Bézout's theorem for curves states that, in general, two algebraic curves of degrees $m$ and $n$ intersect in m·n points and cannot meet in more than $m·n$ points unless they have a ...
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### Parametric Interpolation in the Plane

Given $i+j$ points in the plane, when can we find $x(t),y(t)$, polynomials of degree $i$ and $j$ respectively such that the parametric curve $(x(t),y(t))$ goes through each point? We can do this ...
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### Order and residue of 1-form $x^{-1}dx$

Consider the rational 1-form $x^{−1}dx$ on $\mathbb{P}^1$. I am asked to compute its order and residue at all $P \in \mathbb{P}^1$. Could somebody help me with this? I do not really how to start ...
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### The set of commutative matrices is an irreducible algebraic variety

Let $A, B$ matrices $n \times n$. Let $X = \left\{(A, B) \in \mathbb{A}^{2n^2} \mid AB = BA \right\}$. Prove that $X$ is algebraic and irreducible variety.
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### Non Vanishing Thetanulls

Let $C$ be a smooth curve over $\mathbb C$, a theta characteristic $\kappa$ is said to be non-vanishing if $h^0(\kappa)=0$. Does there exist always a non vanishing theta characteristic?
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### If ${L_1} \subseteq {L_2}$ and and $L_1$ is union of continuous curves and $L_2$be algebraic curve. Can we say that $L_1$ is piecewise $C^∞$ curve?

Let $L_2=\{(x,y):x,y\in R,f(x,y)=0\}$ be algebraic curve, and $f(x,y)$ is polynomial. If ${L_1} \subseteq {L_2}$ and $L_1$ is union of continuous curves. Can we say that $L_1$ is piecewise $C^∞$ ...
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### Can we say that, every algebraic curve is a piecewise ${C^\infty }$ curve?

Let $L$ be algebraic curve. Can we say that, $L$ is a piecewise ${C^\infty }$ curve?
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### Let $\{(x,y):x,y\in R,f(x,y)=0\}$ be algebraic curve, and $f(x,y)$ is polynomial. Why does this algebraic curve, is continuous?

Let $\{(x,y):x,y\in R,f(x,y)=0\}$ be algebraic curve, and $f(x,y)$ is polynomial. Why does this algebraic curve, is continuous?
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### Compute the genus of a curve with a flex point

The genus of a smooth plane curve is $g=\frac{(d-1)(d-2)}{2}$ and I know that if the curve has $n$ nodes the genus decreases by $n$. What happens if the curve has singular (non ordinary) points? In ...
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### Euler characteristic singular surface

The setting is the one of algebraic curves over the complex numbers. It is known that in an irreducible nodal curve each node reduces the arithmetic genus by one: if $\tilde{C} \rightarrow C$ is the ...
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### Projective Mapping of a Sphere

Show that the projective completion of the curve $Y=X^2$ is topologically a sphere. Consider the parametrization $X=t, Y=t^2$, where t ranges the sphere $\mathbb C\cup {\infty}$. How do I prove this ...
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### When is a hypersurface rationally connected?

A projective variety $X$ is said to be "rationally connected" if any two points on it can be connected by a map $\mathbb P^1 \to X$. Let $X$ be a smooth hypersurface in $\mathbb P^n_k$ defined by a ...
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### Smooth completion of algebraic curves

I am having trouble understanding the concept of smooth completions of algebraic curves. I know the definition (smooth, complete curves which contain the curve X as an open subset ) and according to ...
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### local equation of a divisor, localization and local ring at a point

Let $X$ be a regular, proper surface (integral, separated, of fin. type) over a perfect field $k$ and let $Z\subset X$ be an integral irreducible subscheme. Moreover $\eta$ is the generic point of $X$ ...
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### The set of curves of degree $d$ with two singular points or a degenerate singular point is closed?

Suppose $d>0$, $k$ is a field and $L_d$ is the space of projective plane curves over $k$ of degree $d$, namely $L_d=\mathbb P(k[X,Y,Z]_d)$, where $k[X,Y,Z]_d$ is the vector space of homogeneous ...
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### Automorphism of elliptic curve, Vakil 19.10.E

There are many other proofs of finding all the possible automorphism groups of elliptic curves, but I am interested in the $Hint$ and the corresponding proof in the following exercise from Vakil's ...
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### A 3D curve correlation

Forgive me if its too basic, but i am looking to read some materials about a subject in which i don't know its name/field. So what we need to do, is to get a 3 axises curve, with unknown shape, that ...
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### Automorphism group of genus 1 curve

Suppose $E$ is a regular curve of genus 1 over a field $k$ that is not necessarily algebraically closed. The automorphism groups of $E$ forms a one parameter family. If $k=\mathbb{C}$, then $E$ could ...
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### Picture of elliptic curve double covers Rieman sphere

For simplicity, let us assume we live in the field $\mathbb{C}$. Then all elliptic curves are hyperelliptic, so they admit a double cover of $\mathbb{P}^1$ branched over four points, whose preimage ...
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### Automorphism group of genus 2 curve

Suppose C is a genus 2 curve over a field k such that char k is not 2. Is there an easy way to show that the automorphism group is finite? If we assume k is algebraically closed then C is ...
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### compute order x/z in Q

Let $x^3+y^2z+yz^2+\alpha^3z^3=0$ be a curve over $F_{16}$ where $\alpha$ is primitive element for $F_{16}$ and $\alpha^4+\alpha+1=0$. The point $Q=(0:1:0)$ is only infinity point this curve. I want ...