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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1answer
66 views

$L(D)$ is Vector Space

Given a divisor $D$ on a curve $X$, define $L(D)=\{0\}\cup \{f \in k(X),f\ne 0 \, | (f)+D \ge 0\}$. where $(f)=\sum \nu_P(f)P$ and $ \upsilon_{P}(f)= |zeros| − |poles| $ of $f$ at $P$. I want to ...
2
votes
1answer
239 views

Calculating the projective closure with more than one generator

I am given a variety $X = Z(f_1,f_2)$ in affine 3-space (in $x,y,z$), and I would like to compute its projective closure $Y = Z(g_1,\dots,g_n)$ in projective 3-space (in $x,y,z,w$). I have seen this ...
2
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1answer
51 views

Proving that a map is a birational equivalence

I am trying to prove that the map $\phi:P^1\to X = Z(x^2y^3-z^5)$, given by $[r:s]\mapsto [u^5:v^5:u^2v^3]$ is a birational equivalence, i.e. that there exists some map $\psi:X\to P^1$ such that ...
4
votes
2answers
380 views

Coordinate ring of the unit circle is never a UFD?

I'm reading some notes about coordinate rings. On the third example on the second page, the author notes that the coordinate ring $K[\mathcal{C}]$ is not a UFD. If $f=X^2+Y^2-1$, then in ...
2
votes
1answer
92 views

Existence of finite morphism to projective line

As we all know, Belyi's theorem says: A complex curve $X$ is defined over a number field, if and only if there exists a finite morphism $t:X\to \mathbb{P}^1_\mathbb{C}$ of varieties over $\mathbb{C}$ ...
2
votes
1answer
68 views

Existence of $g_{d}^{r}$ implies existence of which $g_{d'}^{r'}$'s?

Suppose I have a Riemann surface with a $g_{d}^{r}$. I am wondering what $g_{d'}^{r'}$'s exist for sure. For instance, since $h^{0}\left(L\left(-p \right)\right)$ is either $h^{0}\left(L\right)$ in ...
0
votes
2answers
68 views

5th order Polynomial not accurate enough?

I have a data plot XY that goes from (X 0-127, Y -70.0 - 6.0 db) Im using the 5th order polynomial function from plotting this data on this site [http://www.zizhujy.com/en-us/Plotter][1] However, ...
2
votes
1answer
152 views

Intersection Multiplicity and Multiplicity of Zeros in Polynomial

I study coding theory and we use the textbook Fundamentals of Error-Correcting Codes . In the section related to Algebraic Geometry Code, we need to compute Intersection Multiplicity of two curve in ...
3
votes
1answer
116 views

Proof of the Belyi's theorem: where it is really used the hypothesis?

Consider the Belyi's theorem: If a smooth projective curve $X$ is defined over $\overline{\mathbb Q}$, then there exists a finite morphism $X\longrightarrow\mathbb P^1(\mathbb C)$ with at most $3$ ...
1
vote
1answer
123 views

When branch points lie in $\mathbb P^1\left(\overline{\mathbb Q}\right)$.

I'm working on the following problem for several days without finding any solution: Let $X$ be a complex smooth projective curve and suppose moreover that $X$ is defined over $\overline{\mathbb ...
2
votes
2answers
70 views

Strange property of parametrization of a class of plane curves

My studies lead me to the following parametrization of perhaps a new class of plane curves ( which are similar in shape to the classical sinusoidal spirals but not identical ). If the curves are not ...
10
votes
3answers
217 views

Is there a better way to find the polynomial equation for this curve?

Consider the curve in $\mathbb{R}^2$ defined by the equation $$ x^{1/3} + y^{1/3} + (xy)^{1/3} = 1, $$ where $x^{1/3}$ denotes the real cube root of $x$, etc. Since the equation above involves only ...
0
votes
1answer
195 views

Find all the intersection points of a vector parabola (in R3) and a sphere

Given that I have a vector in R3 (7t, 10t - 2t^2, 5t) | (These numbers are arbitrary for the sake of the process) A sphere centered at the point ( 15, 25, 10) with a radius of 20 There is a ...
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0answers
57 views

Divisors of Fermat Curves.

I've been studying Algebraic Geometry (in coding theory), and my book has been very ambiguous about what some of these ideas actually look like. So I was curious as to what some of the Divisors of ...
5
votes
1answer
252 views

Exactness of the pullback of the Euler sequence

Let $(L,V)$ be a base point free $g^r_d$ on a curve $C$. Then we have an exact sequence $0\rightarrow M_{L,V}\rightarrow\mathcal{O}_C^{r+1}\xrightarrow{\theta} L\rightarrow 0$, where we just let ...
2
votes
1answer
112 views

Line Meeting a Plane Curve at One Point

Given a curve (smooth, projective, irreducible) $X$ in $\mathbb{CP}^2$, this curve meets all other curves in the same space. Generically, it will meet a line (a copy of $\mathbb{CP}^1$ in ...
9
votes
4answers
313 views

What are some applications of the Weil conjectures for algebraic curves?

I have been interested in the Weil conjectures for some time, and the easiest place to start has been in studying them for elliptic curves. I've been able to see some of their applications and ...
4
votes
1answer
121 views

Degree of ramification divisor and number of fixed points under a group action.

Let $C$ be a projective plane nonsingular complex curve and a finite group $G$ acts on $C$. Consider the quotient $f: C\rightarrow C/G=:C'$. Then, by Riemann–Hurwitz $2g_C-2=|G|(2g_{C'}-2)+\deg R,$ ...
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vote
0answers
104 views

Irreducible Linear Subspace

Let k be an infinite field. Prove that any linear subspace of $A_k^n$ is irreducible. My first question is, what would a linear subspace be? Is is a variety that is generated by linear equations? ...
4
votes
0answers
95 views

On the bidegree of a curve in $\mathbb{P}^1 \times \mathbb{P}^1$

I was reading Beauville's Complex algebraic surfaces, at page 5 there is an example in which curves in $\mathbb{P}^1 \times \mathbb{P}^1$ are classified by the bidegree up to linear equivalence. I'm ...
3
votes
1answer
72 views

Elliptic points on modular curves and a uniformising parameter

I'm working with the modular curves $X_0(p)$ for $p=2,5$, with $$f_2=q\prod_{n\geq1}(1+q^n)^{24}$$ and $$f_5=q\prod_{n\geq1}(1+q^n+q^{2n}+q^{3n}+q^{4n})^6$$ being inverses of the Hauptmodul and ...
3
votes
1answer
119 views

$d$-branched coverings and finite morphisms of degree $d$

Consider a smooth projective curve $X$ over $\mathbb C$ (so $X$ is a projective $\mathbb C$-scheme, integral, of finite type....), and let $t: X\longrightarrow\mathbb P^1_{\mathbb ...
7
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2answers
217 views

Branch points lie in $\mathbb P^1\left(\overline{\mathbb Q}\right)$

Let $X$ be a complex smooth projective curve and suppose moreover that $X$ is defined over $\overline{\mathbb Q}$. Now consider a finite map $f:X\longrightarrow\mathbb P^1(\mathbb C)$ of degree $d$ ...
3
votes
1answer
486 views

Picard group and jacobian of a singular curve

I found somewhere written that the jacobian of a reducible curve is the product of the jacobians, which seems quite reasonable to me. Where can I find some proof of it and a description of the Picard ...
3
votes
1answer
61 views

Valuation rings and total order

Let $K$ be a field and $\mathcal{O}$ be a valuation ring of $K$ (So $\mathcal{O}\subset K$ is an integral domain with the property that either $x$ or $x^{-1}$ is in $\mathcal{O}$ for all $x\in K$). ...
2
votes
0answers
50 views

Zeta Function of a Curve

In general, is there a simple way of computing the zeta function of a curve (or variety) over $\mathbb{F}_q$? Here $q$ is an odd prime power. I've seen a nice computation for both affine and ...
11
votes
3answers
554 views

Applications of Belyi's theorem

Belyi's theorem (1979) is stated as follows: A smooth projective curve over $\mathbb C$ is defined over a number field if and only if there exists a finite morphism (of varieties over $\mathbb ...
2
votes
1answer
326 views

projective non-singular curve

I am working on algebraic curves at the moment and I can not find a proper definition of the projective non-singular curves. My goal is understand that the category of non-singular projective curves ...
3
votes
1answer
119 views

Max Noether's fundamental theorem aplication

Let $C$ be a irreducible cubic in the projective plane and let $F,F^\prime$ be two algebraic curves of degree $m$ satisfying $(C,F)=\Sigma_{i=1}^{3m}p_i$ and $(C,F^\prime)=\Sigma_{i=1}^{3m-1}p_i+q$, ...
3
votes
1answer
251 views

What is a field of definition of a morphism?

An algebraic variety $X$ over a field $K$ is a $K$-scheme integral separated and of finite type over $K$. Definition: If $L$ is a subfield of $K$ we say that $X$ is defined over $L$ if there ...
6
votes
2answers
774 views

Geometric interpretation and computation of the Normal bundle

My question concern the definition, geometric meaning, and usage of the normal bundle in algebraic geometry. Let $X$ be a nonsingular variety over an algebraically closed field $k$, and $Y\subseteq ...
5
votes
1answer
110 views

How are Jacobians of genus $3$ curves different from one another?

There are two types of smooth projective (complex) curves of genus $3$: plane quartics, and hyperelliptic curves. The Torelli morphism $M_3\to A_3$, assigning a curve to its (principally polarized) ...
4
votes
2answers
92 views

Degree 2 Fermat curve

I'm trying to solve the following exercise: Prove that the variety $V\subset \mathbb{CP}^2$ defined by $x^2+y^2+z^2=0$ is isomorphic to $\mathbb{CP}^1$. What I've done: I tried to define an explicit ...
2
votes
1answer
100 views

Is $y^2 =$ quartic in $x$ smooth at infinity?

Let $q(x)\in K(x)$ be a quartic polynomial in x with distinct roots over the algebraically closed field $K$. Consider the curve $C\subset \Bbb P^2$ given by $y^2-q(x)$. Is $C$ smooth? Well, at least ...
4
votes
2answers
282 views

Prym variety associated to an étale cover of degree 2 of an hyperelliptic curve.

In view of this question, I have an additional question. The situation is as follows. Let $C$ be the hyperelliptic curve over $\mathbb{C}$, which is given on an affine by the equation $y^2 = x^5 +1 ...
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vote
0answers
80 views

Elliptic curves in $\Bbb P^3$

How can I check that a curve inside of $\Bbb P^3$ is an elliptic curve? Specifically, let $C$ be the plane cubic $$C:aX^3+bY^3+cZ^3=0$$ and $\phi:\Bbb P^2\to \Bbb P^3$ given by ...
2
votes
1answer
97 views

Endomorphisms of the projective line

Let $f:\mathbb{P}^1 \to \mathbb{P}^1$ be a degree 1 endomorphism of the the projective line over $\mathbb{C}$. It is well known that $f$ is an automorphism, and moreover it is determined by its value ...
6
votes
1answer
128 views

Nonsingular projective variety of degree $d$

For each $d>0$ and $p=0$ or $p$ prime find a nonsingular curve in $\mathbb{P}^{2}$ of degree $d$. I'm very close just stuck on one small case. If $p\nmid d$ then $x^{d}+y^{d}+z^{d}$ works. If ...
5
votes
1answer
282 views

How to compute this Riemann surface?

This question is related to other more general question that I asked Computing Riemann surfaces of a given algebraic function. By the way, I've found an approaching in Markushevich's book that ...
1
vote
0answers
57 views

Example of a Regular Map

I am working with Shafarevich's "Basic Algebraic Geometry 1". Example 1.15: The map $f(t)=(t^2,t^3)$ is a regular map on the line $\mathbb{A}^1$ to the curve given by $y^2=x^3$. I am not ...
6
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2answers
214 views

References for elliptic curves over schemes

As in the title, I want some references about theories for elliptic curves over rings(not fields) or over schemes. I heard that behaviours(?) of such elliptic curves are not as simple as elliptic ...
2
votes
1answer
65 views

Does $\,f_* \mathcal{O}_{X_T} \cong \mathcal{O}_{T}$ hold in this situation?

Let $X$ be a scheme over $S$ and consider the following hypothesis : \begin{cases} \; (1) \quad f:X\to S \text{ is quasi-compact and quasi-separated } \\\\ \; (2) \quad f:X\to S \text{ admits a ...
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0answers
68 views

Pushforward of differentials (?) and Abel Jacobi map on principal divisors

Let $X$ be a complete one-dimensional curve over $\Bbb C$ (please add any hypotheses in case I forgot them). Then we have the Abel-Jacobi map $$\mathcal{AJ}: \text{Div}_0 \to \Bbb C^g/ \Lambda$$ ...
2
votes
0answers
47 views

Generalisation of circumscribed circle for polygones instead of triangles.

Given a triangle, there exists a unique circle passing through its $3$ vertices. I wonder if more generally for any $n\geq3$ we can define a family $\mathcal{C}_n$ of convex closed curves in the ...
6
votes
1answer
171 views

Local parameter of curves in affine n-space

I'm looking for a double answer to this question: a mathematical one (say, if the statement is correct or not) and a philosophical one (say, why we do expect this to be true, or not). Let $k$ be a ...
3
votes
1answer
48 views

Curve with acnodes over closed fields?

From Wikipedia: An acnode is an isolated point not on a curve, but whose coordinates satisfy the equation of the curve. The term "isolated point" or "hermit point" is an equivalent term. I was ...
4
votes
0answers
146 views

Computing Riemann surfaces of a given algebraic function

I've never seen written in a book a way or an algorithm for computing Riemann surfaces of a given algebraic function. I would like to know how to construct such Riemann surface using intuitive cutting ...
3
votes
4answers
327 views

how to find the equation of one curve in XY plan which satisfies such functions?

I want to find the equation of one curve in $X-Y$ plan which satisfies the functions as follows: 1) $A(x_1,y_1)$, $B(x_2,y_2)$ are two known points and $f(x,y)$ (or $y(x)$) is the equation of the ...
7
votes
1answer
230 views

Exercise 1.11 of Eisenbud

I'm doing the exercises from Eisenbud's Commutative Algebra with a view toward Algebraic Geometry, and I don't understand part of one of them, ex. 1.11 a): Exercise 1.11 a: Over $\mathbb{C}$, ...
5
votes
1answer
286 views

Pullback of an ample line bundle through a projection

Assume $X_1$ and $X_2$ are two smooth projective curves and let $M_i$ be an ample line bundle on $X_i$, for $i=1,2$. Further, denote the natural projection map on the $i$-th factor by ...