# 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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### Endomorphism ring of Drinfeld modules.

Let $\mathcal{X}$ be a smooth geometrically irreducible projective curve over $\mathbb{F}_q$. Fix a closed point $\infty\in \mathcal{X}(\bar{\mathbb{F}_q})$. Let $K$ be the function field of ...
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### Smoothing transverse self-intersection

Let $S$ be a complex surface, and let $C \subset S$ be an immersed complex curve with a transverse self-intersection at point $P$. Let $\tilde{C}$ be a curve obtained from $C$ by smoothing the ...
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### Which curves have reflexive structure sheaf?

Let $C$ be a curve, i.e. a purely one-dimensional scheme, embedded in a smooth projective threefold $X$. For a coherent sheaf $E$ of codimension $c$ on $X$, let $E^D=\mathscr Ext_X^c(E,\omega_X)$ be ...
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### Cohomology of rational quartic in $\mathbb{P}^3$

I have to do this exercise. Let $X\subseteq\mathbb{P}^3$ a rational curve of degree $4$. Show that $$H^1(\mathcal{O}_X(1))=0=H^1(\mathcal{I}_X(2))$$ I tried to look at $X$ as closed immersion by ...
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### Is there something similar to $\mathbb{R}^2$ for elliptic curve point representation?

Let $E$ be an elliptic curve over a finite field $\mathbb{F}_p$ and denote with $E(\mathbb{F}_p)$ its set of points over $\mathbb{F}_p$. Consider a coordinate system in $\mathbb{R}^2$. Every point is ...
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### The equation of a jelly bean curve?

What is the equation of a curve with jelly bean shape? I have found a quartic equation for bean shaped curves, but nothing for jelly beans. If somebody doesn't know the shape, here is a link: jelly ...
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### Computing cohomology over projective curve in $\mathbf{P}^3$

Let be $k$ an algebraically closed field and Let be $X\subseteq \mathbf{P}^3:=\mathbf{P}_k^3$ a smooth, irreducible curve that is not contained in any hyperplane. Let's call $d=\deg(X)$. A well known ...
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### Prove that the tangent at $P$ intersects $C$ twice at $P$ and once at $4P$; the tangent at $5P$ intersects $C$ twice at $5P$ and once at $2P$.

This is a problem from Conics and Cubics by Bix. Please help me answer this one. Let $C$ be a nonsingular, irreducible cubic with a flex $O$. Add points (commutative) of $C$ with respect to $O$ as ...
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### An application of the Max Noether's theorem

I'm reading chapter IV of Robert J Walker's book 'algebraic curves'. The last section of this chapter is about Max noether's AG+BF theorem. I am stuck on an exercise in this section. The exercise ...
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### Prove that $2P$ is a flex of $C$ collinear with $P$ and $3P$ and $4P$ is a flex of $C$ collinear with $5P$ and $3P$.

I encountered this problem from Conics and Cubics by Bix. Please help me answer this. Let $C$ be a nonsingular, irreducible cubic with a flex $O$. Add points (commutative) of $C$ with respect to $O$ ...
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### Does roots of L-polynomial of curve being roots of unity imply the curve is supersingular?

If $C$ is a supersingular curve over $\mathbb F_q$, then $\frac1{\sqrt q}$ of roots of $L$ function are roots of unity. What about converse? If $\frac{1}{\sqrt q}$ of roots of $L$ polynomial are ...
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### How can I describe the intersection between a circle and a curve?

I have a curve C and a point x in the curve. At the point x, I draw a circle B with radius r and centered at point x. That circle B will segment/intersect (with) the curve C as red sub-curve line. I ...
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### Resultants on Projective Curves

The two curves $$F=(X^2+Y^2)^2+3X^2YZ-Y^3Z$$ and $$G=(X^2+Y^2)^3-4X^2Y^2Z^2$$ on $P^2(\mathbb{C})$contain the point $(0:0:1)$ as their point of intersection. Therefore, the resultant with respect to ...
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### does the $\zeta_K$ function of a function field determine the genus of that function field?

Let $K_1$ and $K_2$ both be function fields over a finite field (or algebraic curves, if you like) with zeta functions $\zeta_{K_1}$ and $\zeta_{K_2}$. Say that $\zeta_{K_1} = \zeta_{K_2}$ - so that ...
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### Rational Parametrization of Projective Curves

I wish to show that the curve on $P^2(\mathbb{C})$ given by $$F(X,Y,Z)=(X^2-Z^2)^2-Y^2(2YZ+3Z^2)$$ is a rational curve. I tried to do a quadratic transformation by determining ...
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### Method of finding Arc length parameterization of a 3d curve

r(t) = cos^3 t i + sin^3 t j; 0 < t < pi/2. r'(t) = -3cos^2 t sin t + 3 sin^2 t cos t ||r'(t)|| = 3 sin t cos t Now to find the arc length parameterization, we need S = integration from t0 ...
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### Rational maps between affine varieties

If I want to check that the map $$\phi:C_1\rightarrow C_1,\hspace{0.5cm}\phi(x,y)=(\phi_1(x,y),\phi_2(x,y))$$ between two affine plane curves is rational I just should check that $\phi_1$ and ...
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### Rationality of the Lemniscate.

This question is exercise 2 of Chapter 4 in Kunz' textbook of algebraic curves. Let $f$ be the lemniscate with equation $$(X^2 + Y^2 )^2 = α(X^2 − Y^2) \;\; (\alpha \in K^\times )$$ and let ...
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### Is every Riemann surface a 2-sheeted covering?

Given an algebraic curve $X$ over $\mathbb{C}$, i.e. a Riemann surface and a fixed set of pairs of points $S=\{(p_1,q_1),...,(p_1,q_1)\}$ is there an algebraic curve Y, possibly singular, and a map ...
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### Looking for the equation or algorithm for a mystery dataset [closed]

I'm a programmer by trade, although I did both A-level and engineering maths at University, I'm a little rusty. I'm trying to reverse engineer a pretty shoddy bit of legacy code. I have two sets of ...
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### Gluing together Riemann surfaces, don't see why $Z$ is union of two compact sets.

Consider Lemma 1.7 from page 60-61 of Miranda's Algebraic Curves and Riemann Surfaces. For a link to the book, see here. Lemma 1.7. With the above construction, $Z$ is a compact surface of genus ...
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### Cohomology and normalization of a curve

Suppose $C$ is a projective curve such that $H^1(C,\mathcal{O}_C)=0$ and let $\pi:\tilde{C}\longrightarrow C$ be the normalization morphism. How can I show that ...
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### Intersection of following pair of parabolas at infinity? [closed]

What is the intersection multiplicity of the following pair of parabolas at infinity:$$y = x^2,\text{ }y = x^2 + 1?$$
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### Maps induced by divisors and degree

I feel very stupid to ask this question but I've serached everywhere and I've not found a clear (to me) answer. Is a well known result in Curve theory (over $k=\bar{k}$) that a Divisor $D$ on a curve ...
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### The Hirzebruch surface $\mathbb F_n$ contains a unique irreducible curve with negative self-intersection.

Consider the $n$-th Hirzebruch surface $\mathbb F_n:=\mathbb P_{\mathbb P^1}(\mathcal O_{\mathbb P^1}\otimes\mathcal O_{\mathbb P^1} (n))$, and let $C_0\subseteq \mathbb F_n$ a section of the ruling ...
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### C: a smooth projective curve/k. $x\in\bar{k}(C).dx$ is a basis for $\Omega_C,\Rightarrow\bar{k}(C)$ is a finite separable extension of $\bar{k}(x)$.

Let C be a smooth projective curve over a field $k$ and $x\in\bar{k}(C)$. If $dx$ is a basis for $\Omega_C$, then $\bar{k}(C)$ is a finite separable extension of $\bar{k}(x)$.
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### $\mathbb P^1\times \mathbb P^1$ is minimal

I'm trying to prove that the complex surface $\mathbb P^1\times \mathbb P^1$ is minimal. I'd like to prove it directly, namely by showing that there are not exceptional curves. Suppose that $E$ is a ...
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