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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32 views

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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38 views

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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48 views

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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1answer
88 views

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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2answers
72 views

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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54 views

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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12 views

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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11 views

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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17 views

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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1answer
33 views

Show that the irreducible quartic projective curve is rational

How can I show that the irreducible quartic curve $\Gamma=V_+((x^2-z^2)^2-y^2(2yz+3z^2))$ on $P^2(\mathbb{C})$ is rational by considering the family of conics through the double points $(1:0:1), ...
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24 views

Solving the curve equation for logarithmic decay using two anchor points.

I would like to have an adaptable logarithmic curve equation that I can then find y for any value of x. I have two points (x1,y1) and (x2,y2). My data requires constant decay (financial discounting ...
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40 views

Relationship between discriminants and smoothness of curves

My understanding of the use of the discriminant in elliptic curve theory is to test whether an elliptic curve in Weierstrass normal form over a field not of characteristic either 2 or 3, $y^{2} = ...
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67 views

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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27 views

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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29 views

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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4 views

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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25 views

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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1answer
18 views

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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41 views

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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48 views

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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78 views

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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49 views

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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32 views

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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1answer
54 views

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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1answer
56 views

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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29 views

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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48 views

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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6 views

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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1answer
53 views

$\mathbb P^1\times \mathbb P^1$ 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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1answer
65 views

Degree of $f:\mathbb{P}^1_k\rightarrow \mathbb{P}^1_k$

Let $k$ be an algebrically closed field and consider $\mathbb{P}^1_k$ the $1$-dimensional projective space over $k$. My question is the following: Let consider $f:\mathbb{P}^1_k\rightarrow ...
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1answer
54 views

Geometrically ruled surface, sections and intersection numbers.

Consider a geometrically ruled surface $\pi: S\longrightarrow C$ where both $S$ and $C$ are projective, non-singular and complex ($C$ is obviously a curve). By using Tsen theorem one can show that ...
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49 views

Tricks to find $\dim H^0(O(D)(n))$

Let $X$ be a curve of genus $g\neq 0$ and $D$ a divisor on $X$. If $n \in \mathbb{Z}$ and $O(D)(n)=O(D)\otimes O(n)$ is the twist of $O(D)$ by $n$, there is a manageable proceedings to find the ...
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151 views

Understanding the connection between the projective space and the affine plane

Suppose we have a point $P=[x,y,z]\in \mathbb P^2$. Then at least one of the coordinates is not zero. Suppose $z\neq 0$. So we have write $P$ as $[x/z,y/z,1]$ and this point belongs to $(x/z,y/z)$ ...
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1answer
45 views

Looking for $\dim _{K}(m_{(0,0,0)}/m_{(0,0,0)}^{2})$ for certain algebraic variety.

Let $X=V(X_{2}^{2}-X_{0}^{2}X_{1},X_{1}^{3}-X_{0}^{4},X_{0}^{3}-X_{1}X_{2},X_{1}^{2}-X_{0}X_{2})\subseteq\mathbb{A}^{3}_{K}$. We denote $$ m_{(0,0,0)}=\{\overline{f}\in K[X]:f(0,0,0)=0\}, $$ where ...
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1answer
61 views

Logistic curve through three points

I need to find a logistic curve that passes through three points exactly. This means I cannot do a best fit but rather must use simultaneous equations. Essentially this is used to model population ...
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15 views

Is a multiple of a hyperelliptic curve hyperelliptic?

Let $C$ be a curve of genus 2 over $\mathbb{C}$. So $C$ is hyperelliptic, that is it admits a degree 2 map to $\mathbb{P}^1$. Is a power of $C$ say $nC$ hyperelliptic too, $n\geq 2$? It is not of ...
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How to prove $l(D)-l(-2(\infty)-D)=\text{deg}D-1$ without using Riemann-Roch theorem?

How to prove $l(D)-l(-2(\infty)-D)=\text{deg}D-1$ without using Riemann-Roch theorem, where $D$ is a divisor on $\mathbb{P}^1$?
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Explicit computation of Serre duality

Given a projective non-singular curve $X$, the a Serre duality asserts an isomorphism between $H^0(X,\Omega^1_X)$ and dual of $H^1(X,\mathcal{O}_X)$. My question is how to compute the dual elements in ...
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1answer
55 views

“path-connectedness” of an algebraic variety

Let $X$ be an irreducible algebraic variety over a field (supposed to be algebraically closed if necessary). How to proove that any two closed points of $X$ can be connected by a finite number of ...
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24 views

Establishing Linear Equivalence of Divisors on Curves

I am trying to do some questions from Geometry of Algebraic Curves by Arbarello-Cornalba-Griffiths-Harris. Here are some of the examples: Exercise A3: Curve: $y^2=x^3+1$. Let $\Gamma=C$ be the ...
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1answer
85 views

Looking for an affine curve not isomorphic to an affine plane curve.

I want to find an affine curve not isomorphic to an affine plane curve (as simple as possible). I am trying to find an affine curve $X\subseteq\mathbb{A}^{n}_{k}$ such that its coordinate ring is not ...
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56 views

Stalks of invertible sheaves on curves

I have just found out that I have maybe not understood very well what an invertible sheaf looks like. Let $X$ be a (regular, integral, separated, whatever you want) curve and $\mathcal{L}$ an ...
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140 views

About a theorem in Beauville's book.

Look at the following theorem (due to Castelnuovo) taken from Beauville's book on complex surfaces: I have a question about the behavior of $f$ on the exceptional curves generated by $\eta$. ...
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1answer
75 views

Question on the proof of Hartshorne IV.3.7

Proposition 3.7 in Hartshorne's Algebraic Geometry is the following. Let $X$ be a curve in $\mathbb P^3$, let $O$ be a point not on $X$, and let $\phi: X \rightarrow \mathbb P^2$ be the morphism ...
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12 views

Topology of the cuspoidal cubic

Let $C$ denote the set of solutions to $zy^2 = x^3$ inside of $\mathbb{C}P^2$. Someone told me that this space is homeomorphic to the pinched torus (or pinched sphere depending on how you pinch) - ...
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23 views

Morphism of ringed topological spaces

Let $f:X\rightarrow Y$ a morphism of ringed topological spaces. Let $V$ be an open subset of $Y$ containing $f(X)$. I want to show that there exists a unique morphism $g:X\rightarrow V$ whose ...
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11 views

The relationship between ramification index and “degree of maps between algebraic curves”

This problem is Exercise II.2.3. in Silverman's 'Arithmetic of elliptic curves' . Consider in the field $\mathbb{C}$. We need to verify directly that \begin{equation} \sum_{P\in ...
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1answer
27 views

How can I figure out what the log function being used based off the X and Y values?

I have a chart where Microsoft .NET has automatically scaled it using a (supposedly) log10 function of some kind. I need to figure out what formula they're using for the value at each tick mark. The ...
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1answer
13 views

Do problem weights change as the overall grade of an assignment is curved?

When I get papers back for class, there's often a question or two that I know I could make a case for getting credit for my answer, but ultimately decide it's not worth the extra % in the grand scheme ...