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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30 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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6 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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28 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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29 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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46 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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55 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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1answer
81 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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54 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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35 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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69 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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59 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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37 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. A well known result in Curve theory (over $k=\bar{k}$) states that a divisor $D$ on a ...
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56 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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7 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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55 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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69 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
64 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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50 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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159 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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46 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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71 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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12 views

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 $\mathbb{P}^1$?
3
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66 views

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
67 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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28 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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94 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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68 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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149 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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82 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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15 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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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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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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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 ...
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29 views

Is this parametric equation describe a circle?

Let $w=\varepsilon\beta(t)-i\sqrt{\beta(t)^2-1}$, where $\beta(t)=\cosh t$ and $\varepsilon >0$. the parametric function is defined as $x+iy=\frac{2w}{|w|^2+1}$ and $z=\frac{|w|^2-1}{|w|^2+1}$. ...
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1answer
54 views

Example of two subvarieties of $\mathbb{P}^2$ that are isomorphic but not projectively equivalent.

Two curves $C_1$, $C_2 \subset \mathbb{P}^2$ are called projectively equivalent if there is a projective change of coordinates $\phi: \mathbb{P}^2 \to \mathbb{P}^2$ so that $\phi(C_1) = C_2$. What is ...
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1answer
50 views

Relation between curves on a surface and divisors

Given a projective surface $S$, the irreducible curves contained in $S$ are exactly, by definition, the prime (Weil) divisors of $S$. I was wondering what are reducible curves on $S$ in terms of ...
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75 views

Corollary of Riemann Roch theorem

I'm studying about Riemann Roch theorem for curves and its consequences (in algebaic gemoetry). In Hartshorne I've found an exercise equivalent to this assert: Let $D$ be an effective divisor of ...
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16 views

Any higher level maths or theories for epicycloids and/or hypocycloids?

For my 12 grade folio task on cycloids, I need to research hypocycloids and/or epicycloids. I need to consider: - exploring how the relative radii of the circles relate to the path - develop ...
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42 views

Degree of pull-back of locally free sheaves under normalization

Let $X$ be a projective, nodal curve, $\pi:\tilde{X} \to X$ be its normalization and $\mathcal{L}$ be an invertible sheaf on $X$. The question is: Is $\deg(\pi^*\mathcal{L})=\deg(\mathcal{L})$? As ...
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42 views

Linking regularity of ideal sheaf with Fitting ideals sheaf

I'm reading Eisenbud's book The geometry of syzygies and I'm quite struck undestanding the argument proposed in Chapter 5, in the section named "Fitting ideals". Remember that a coherent sheaf ...
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1answer
26 views

If for any external point, exactly two tangents can be drawn to an algebraic curve, must the curve be a conic?

Yesterday, my teacher, while proving Poncelet's theorem, seemed to use the fact that if from any external point (external meaning, I assume $f(x,y)>0$ where $f$ is the polynomial of two variables ...
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1answer
57 views

Can a polynomial of degree 2 vanish on three different lines?

Suppose $p(x,y,z)$ is a homogeneous polynomial of degree 2. My question is: Can I have three distinct lines $L_1,L_2,L_3$ in the projective space $\mathbb{P}^2$ such that $p$ vanishes on every ...
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30 views

Self Intersection Formula

Let $\pi$: $X\to \Delta$ be a good degeneration of surfaces with degenerate fiber $X_0 = V_1 + ... + V_n$. Let $C$ be a component of the double curve $V_1 \cap V_2$ I am trying to understand why the ...
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45 views

Dimension of homogeneous polynomials passing through 4 points

Can anyone help me solve the following exercise? Let $p_1,p_2,p_3,p_4$ be distinct points in the projective plane $P^2$. What is the dimension of the vector space of homogeneous polynomials ...
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59 views

bases of a function field

I read this example Example 3.33: Consider the Hermitian curve with equation $x^{r+1}=y^r+y$ over the field $\mathbb{F}_{r^2}$. Here $Q = (0:1:0)$ and $X = x/z$, $Y=y/z$ is a monomial generating ...
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1answer
62 views

There exists a divisor linearly equivalent to $P$ not containing $P$

Let $X$ be a non-singular projective curve over $\mathbb C$ and let $P\in X$ be a closed point. Which is, in your opinion, the easiest way to prove that there exists a divisor $D$ of $X$ satisfying ...
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1answer
90 views

Every point lies on a unique secant through $C$

Let $C \subset \mathbb{P}^3$ be the twisted cubic (i.e., $C=\{(X_0^3:X_0^2 X_1:X_0 X_1^2:X_1 ^3) : (X_0,X_1) \in \mathbb{P}^1\}$). I need to show that every point $Q \in \mathbb{P}^3 \setminus C$ lies ...