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
188 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 ...
4
votes
1answer
244 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
107 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
303 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
119 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 $2g_C-2=|G|(2g_{C'}-2)+\deg R,$ ...
1
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0answers
102 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
92 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
71 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
114 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 ...
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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
474 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
60 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
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3answers
541 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
313 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
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1answer
114 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
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1answer
248 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
701 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
91 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
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2answers
277 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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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
94 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
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1answer
126 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
277 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
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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
208 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
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1answer
64 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 ...
1
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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
167 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
323 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
228 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
274 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 ...
69
votes
0answers
1k views

Geometric interpretation of the Riemann-Roch for curves

Let $X$ be a smooth projective curve of genus $g\geq2$ over an algebraically closed field $k$ and denote by $K$ a canonical divisor. I have some clues about the geometrical interpretation of the ...
4
votes
2answers
64 views

Given $\omega_i \in \Omega_X(U_i)$ can I find $f\in {\cal O}_X(\cap U_i)$ so that $df = \omega_1 - \omega_2$

As per this question: Duality in algebraic de Rham cohomology I am trying to show that the map $H^1(X,\Omega_X) \rightarrow H^2_{\text dR}(X/k)$, where $X$ is a projective algebraic curve over an ...
3
votes
1answer
58 views

Help in this question in Fulton's algebraic curves

I'm trying to solve this question: In item (a) I used the fact $O_a(V)$ is a Noetherian local ring and the only maximal ideal is $(x-a)$. First note that the non-units of $O_a(V)$ are the elements ...
4
votes
1answer
71 views

What is this cycle on the Jacobian of a curve?

Let $C$ be a curve and $J$ it's Jacobian. There is the standard Abel-Jacobi map $a:C\rightarrow J$ which is given by $Q\rightarrow Q-P$ for some fixed point $P$ (here I am regarding $J$ as the degree ...
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votes
1answer
23 views

which regression is better

suppose that we have two input vector and the variables in each vectors are independent and uncorrelated from each other,just only there is relationship between two vector,but not itself in ...
1
vote
1answer
67 views

Why is this is an equivalence relation?

Fulton makes the following definitions: After he defines an equivalence relation: The definitions he made seems very obscure to me and if anyone could show why this relation is an equivalence ...
8
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1answer
450 views

What is normal crossing?

I could not find any reference for normal crossings. The definition here is not so clear to me. In some texts, they sometimes said that two varieties have normal-crossing (non-normal crossing) with ...
3
votes
1answer
70 views

Why there is a minimal element of this set

I'm trying to understand this proof: I know intuitively, but Why formally there is such a minimal element? I need help Thanks
2
votes
1answer
92 views

Vanishing of $R^1f_*\mathcal O_X$

I am probably missing something obvious here, but none the less, here goes: Is the following statement (or perhaps some minor modification of it) true and if so, why: $R^1f_*\mathcal O_X = 0$ for a ...
4
votes
1answer
130 views

Smoothness of the Picard group of a smooth curve

Let $X$ be a smooth projective curve over $k=\bar{k}$ and denote its Picard group by $\operatorname{Pic}(X)$, with the usual scheme structure coming from the representability of the relative Picard ...
1
vote
1answer
178 views

Prove that a curve in P^n of degree n not contained in a hyperplane is rational

The set up is as stated above. We have a projective curve $X$ of degree n embedded in $\mathbb{P^n}$, which is not contained in any hyperplane. We claim that it is therefore rational. The way I have ...
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0answers
18 views

Easy question about coordinate rings

Let $C=V(x^2+y^2-1)$ be an affine algebraic curve. In an online course the professor said $\varphi=\frac{x-1}{y}\in A(C)$, but he didn't explain why. I would like to know how he gets this function ...