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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67 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 ...
3
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51 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 ...
3
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2answers
165 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
50 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 $K[...
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1answer
86 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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0answers
18 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 $\mathbb{P}^1$?
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70 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
86 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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30 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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95 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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71 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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2answers
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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1answer
91 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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17 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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30 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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18 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 \phi^{-1}(Q)}e_{\phi}(...
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1answer
31 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
14 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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1answer
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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votes
1answer
56 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
53 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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2answers
84 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
19 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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0answers
46 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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0answers
47 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
30 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
58 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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32 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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48 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 $f(x_0,...
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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
63 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
93 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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vote
1answer
64 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 $X_{\eta_Y}=f^{-1}(\...
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0answers
15 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 = 4[\wp(z)]^3-g_2\wp(z)-g_3,...
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0answers
35 views

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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1answer
52 views

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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votes
1answer
51 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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0answers
16 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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34 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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34 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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1answer
75 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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votes
1answer
18 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 + t^{-1})...
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votes
1answer
112 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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votes
1answer
93 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 $B$ ...
3
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1answer
40 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 I've ...
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0answers
35 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
40 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 \{(0,...,...
2
votes
1answer
59 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
votes
0answers
50 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 $I$?