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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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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13 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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24 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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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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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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51 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
47 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 ...
2
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2answers
61 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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1answer
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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34 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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35 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
20 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
55 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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28 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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42 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
57 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
80 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 ...
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1answer
57 views

Generic point and pull back

Let's work over $\mathbb C$, Consider a finite morphism between two integral curves $f: X\rightarrow Y$, let $\eta_Y$ (resp $\eta_X$) be the generic points Questions: What is ...
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11 views

Can the Klein Quartic be parameterized by meromorphic upper-half plane functions?

It is known that Elliptic Curves in canonical form can be parameterized by the Weierstrass elliptic function and its derivative on a suitably chosen lattice: $$[\wp'(z)]^2 = ...
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Basepoint free line bundles in $Pic^d(C)$ is an open set?

Let $C$ be a smooth projective curve over $\mathbb{C}$. Let $Pic^d(C)$ denote line bundles on $C$ with degree $d$. Is the subset of basepoint free line bundles an open set of $Pic^d(C)$? Many thanks!
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Simple question from Kunz's book Introduction to Plane Algebraic Curves

I'd like to ask a question that arose when I was reading the book Introduction to Plane Algebraic Curves by Kunz. In the following passage, how can we conclude that ker$\alpha = (q)$? I can see that ...
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1answer
45 views

Flat families and section of family of curves

Let $X$ be a smooth projective surface. Let $L$ be an ample line bundle on $X$. Consider $Y=\{(x,C) : C\in |L|, x\in C\}\subset X\times |L|$. Then $p_2:Y\longrightarrow |L|$ is a family of curves over ...
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13 views

Compute $\ell(W+P)$ and $\ell(W-P)$ for a place $P\in\mathbb{P}(F/K)$ of degree $1$ and any $W$ canonical divisor.

Let $F/K$ be a function field of genus $g$. Let $P\in\mathbb{P}(F/K)$ be a place of degree $1$ and $W$ be any canonical divisor. Determine $\ell(W+P)$ and $\ell(W-P)$. If $\ell(P)=\deg(P)+1$, then we ...
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33 views

a coordinate change of a quadric in P^2 to get one of 3 curves

I need to show that for a quadric curve in P^2(k) of degree 2,if the characteristic of k is not 2 ,then up to a coordinate change any quadric is one of the following :1) non-singular quadric 2) a pair ...
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26 views

Hurwitz bound is not sharp for curves of genus 4.

I'm trying to prove that for any curve $C$ over complex numbers $\mathbb C$ of genus 4 the number of automorphisms is strickly less then $252$. I proved this inequality for hyperelliptic ones. As for ...
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66 views

The translation map between elliptic curves is a rational map

I want to see a reference or a prove that the following map is a rational map: Let E be an elliptic curve,$P\in E$ and $T_p$ defined as $T_p:E\rightarrow E,\text{ }T_P(Q)=P+Q$. It is important ...
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1answer
16 views

Reverse-engineering a parametrization

Let's say you have a polynomial depending on complex parameters $A,B$: $$p(x,y) = (y + Ax)(y + Bx) + (A-B)^2$$ One parametrization of zero points of this polynomial is given by $$ x(t) = -(t + ...
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1answer
73 views

polynomial curve fitting: higher order models' root mean square error does not decrease

I am trying to fit a curve for 15 data points. I started by creating a linear model and observing the root mean square difference, followed by quadratic, cubic and increasing the degree of polynomial ...
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1answer
66 views

Dimension of an affine cone without one variable is equal to the dimension of the projective algebraic set

Let $A:=V(F_1,...,F_k)\subset\mathbb{P}^n$ with $F_j\in k[X_0,...,X_n]$, a projective algebraic set. Let $C(A)\subset \mathbb{A}^{n+1}$ the affine cone over $X$. Show that $\dim A=\dim B$, where ...
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1answer
26 views

Prove that a general monomial curve is smooth

Let $k$ be a field, $n_1<n_2<\cdots<n_r$ positive integers, and $C:=\{(t^{n_1},...,t^{n_r})\mid t\in k\}\subset \mathbb{A}^r$. Show that $C$ is a smooth curve iff $n_1=1$. This is what ...
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27 views

Irreducible components of a cone

I have the following definition for an affine cone: Let $Y\subset \mathbb{P}^n$ be a projective algebraic set, we define the affine cone associated to $Y$ as $C(Y)=\theta^{-1}(Y)\cup \{(0,...,0)\}$, ...
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1answer
27 views

Every irreducible component of an affine cone contains its vertex

Let $X=V(F_1,...,F_k)\subset \mathbb{P}^n$with $F_i\in k[X_0,...,X_n]$ an projective algebraic set. Let $C(X)\in \mathbb{A}^{n+1}$ the affine cone over $X$, that is $C(X)=\theta^{-1}(X)\cup ...
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1answer
54 views

How do I show that there is no irreducible algebraic set $Y \subsetneq \mathbb{A}^n$ such that $Y \supsetneq X$?

Let $f \in k[x_1, \dots, x_n]$ be irreducible. The variety $X = V(f)$ is called an irreducible hypersurface in $\mathbb{A}^n$. How do I show that there is no irreducible algebraic set $Y \subsetneq ...
2
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46 views

Use of Hilbert basis theorem to find a lone generator for $I$.

Let $I \subset \mathbb{C}[x]$ be the ideal generated by $\{x^n + n: n \in \mathbb{N}\}$. How do I use the proof of the Hilbert basis theorem to find a single generator for $I$?
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1answer
87 views

Global sections when we tensor by a degree zero line bundle

Let $C$ be a smooth projective curve over $\mathbb{C}$. Let $A$ be a degree $d$ line bundle on $C$, and $M$ be a degree 0 line bundle on $C$ such that $M^2=\mathcal{O}_C$, that is, it is a 2-torsion ...
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35 views

Is the given set an open subset of the $\mathcal{G}^r_d(|L|)$

Let $X$ be a smooth projective surface over $\mathbb{C}$. Let $L$ be a very ample line bundle on $X$. We have a variety $\mathcal{G}^r_d(|L|_s)$ associated to the linear system of curves $|L|$. The ...
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1answer
46 views

About the degree of a tensor power of a line bundle on a curve

Suppose that $H$ is a hyperplane in some $n$ dimensional complex projective space and $C$ is a smooth curve of positive genus. Can I say that $\deg((H|_C)^{\otimes 2}) = 2\deg(H|_C)$?
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Find polynomial equation for a cardioid in $\mathbb{R}^2$

We have the cardioid with equations: $$x(\theta)=\cos\theta+\frac{1}{2}\cos(2\theta)$$ $$y(\theta)=\sin\theta+\frac{1}{2}\sin(2\theta)$$ I have to show that you can define this cardioid with a ...
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1answer
70 views

Are elliptic curves algebraic varieties?

I got a short question. Are elliptic cubes also algebraic varietes? Say we have $E:y^2=x^3+5x=:f(x)$ Then we can $f(x)=x(x^2+5)$ So it can't be an algebraic variety.. I feel like I am totally ...
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42 views

Solving Integration problem, area of a curve?

I am working my way through a self study book of scientific and engineering principles. Although it covers a pretty broad range of subjects, there are obviously a few maths related topics in it as ...
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1answer
50 views

Rational maps between elliptic curves

I dont understand the definition of rational maps. Here is the definition: Let $E_1$ and $E_2$ be elliptic curves over a field $K$. (projectively written). A rational map $\Phi:E_1\rightarrow E_2$ ...
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31 views

Difference between $ X $ and $ X( \mathbb{C}) $ in a special case.

I would like to know why is, in the case of $ X $ is a smooth curve over $ \mathbb{C} $, $ X( \mathbb{C} ) $ a Riemann surface ? Thank you in advance for your help.
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34 views

Algebraic curves, intersection

Given the two plane curves, $$F=2X_0^2X_2-4X_0X_1^2+X_0X_1X_2+X_1^2X_2$$ $$G=4X_0^2X_2-4X_0X_1^2+X_0X_1X_2-X_1^2X_2$$ I want to calculate the multiplicity of the intersection at $(1:0:0)$, but I have ...
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55 views

Inflection points on cubic curves

Let's call $\Lambda$ the set of cubics generated by, $$F=2X_0^2X_2-4X_0X_1^2+X_0X_1X_2+X_1^2X_2$$ $$G=4X_0^2X_2-4X_0X_1^2+X_0X_1X_2-X_1^2X_2$$ I'm trying to find all the points that are inflection ...
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1answer
50 views

One point in intersection of conic and cubic, intersection multiplicity?

Let $C \subset \mathbb{P}^2$ be the conic defined by $P(x, y, z) = xz + y^2$,and $D \subset \mathbb{P}^2$ be the cubic defined by $Q(x, y, z) = y^2z - x^3 + xz^2$. Show that there exactly one point $p ...
2
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1answer
42 views

Nonsingular curve has 12 points of inflection. [closed]

Let $C \in \mathbb{P}^2$ be the curve defined by the polynomial$$P(x, y, z) = x^4 + y^4 + z^4.$$ Show that $C$ is nonsingular. Show that $C$ has exactly $12$ points of inflection.
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1answer
70 views

Pullback of globally generated line bundle is globally generated?

Let $f:C\longrightarrow C'$ be a finite morphism of curves. Let $A'$ be a line bundle on $C'$ and let $A$ be it's pullback to $C$. If $A'$ is globally generated, is it true that $A$ is globally ...