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Questions tagged [algebraic-curves]

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 of curves are extensively studied as a basic case in singularity theory. Via algebraic function fields and modular curves they have links to number theory.

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2 votes
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144 views

An alternative description of an holomorphic map associated to a complete linear system

I need an help with an exercise in Miranda's book "Algebraic curves and Riemann surfaces". More precisely is the exercise in Problems V.4 I. Given a Riemann surface $X$ and a divisor $D$ on $X$ ...
0 votes
0 answers
95 views

Question on holomorphic map from compact Riemann surfaces into projective space associated with base point free divisor [duplicate]

I am reading through Rick Miranda's Algebraic Curves and Riemann Surfaces. There is a proposition that is left as an exercise for the readers to check. However, I am not sure how to verify it. $\...
0 votes
0 answers
67 views

Coordinate free description of a map induced by a base-point-free divisor on a compact Riemann surface [duplicate]

Let $X$ be a compact Rieman surface, and $D$ be a divisor on $X$ with $|D|$ base-point-free. Note that the vector space $L(D)$ is finite-dimensional, and the complete linear system $|D|$ is naturally ...
1 vote
0 answers
27 views

Understanding Gauss-Manin connection on Arbarello's book

I am trying to understand the Gauss-Manin conecction in order to understand the definition of the Period Mapping on the Moduli space of algebraic curves of genus $g$ and its extension to the ...
-1 votes
1 answer
52 views

A diophantine equation with no solution in positive integers $x,y$ i.e $(y(y+1)+1)^2+1\neq 100x$

Hi I ask separately a question regarding the question where I sktech a special case of the Brocard-Ramanujan problem : Problem : Let $x,y$ be positive integers shows that : $$(y(y+1)+1)^2+1=100x\...
0 votes
0 answers
27 views

Number of bounded and unbounded components of nonsingular real algebraic curve $y^2 - p(x)$

I would like to know if someone can kindly verify my solution to Problem 22.2 from MIT's online course on geometry and topology in the plane. Let $C = \{(x,y) \in \mathbb{R}^2 : f(x,y) = 0\}$ be a ...
0 votes
1 answer
90 views

Finiteness of the intersection multiplicity of plane algebraic curves

Hello guys i am trying to solve excerise 2.7 page 14 from Gathmann notes https://agag-gathmann.math.rptu.de/class/curves-2023/curves-2023-c2.pdf Definition About (a) : Stuck here.Not sure how to ...
1 vote
0 answers
35 views

If the number of intersection of two conics is an odd number, the quadratic forms are not simultaneous diagonalizable

I'm trying to do Exercise 3.6 at the end of this pdf (in $\Bbb CP^2$): Show that the two quadratic forms $$x^2+y^2-z^2, \quad x^2+y^2-y z$$ cannot be simultaneously diagonalized. Attempt 1: Their ...
1 vote
1 answer
688 views

Doubts on the proof about the classification of irreducible projective conics

There are some details in the proof of corollary 3.12 on Frances Kirwan's book complex algebraic curves that I can't understand. Corollary 3.12 Any irreducible projective conic $C$ in $P_2$ is ...
1 vote
2 answers
337 views

Prove that an isolated point of $C: (f=0)\subset \mathbb{R}^2$ must be a max or min of $f: \mathbb{R}^2 \rightarrow \mathbb{R}$.

Let $f\in \mathbb{R}[x,y]$ and let $C: (f=0)\subset \mathbb{R}^2$; we say that $P\in C$ is isolated if there is an $\epsilon >0$ such that $C\cap B(P,\epsilon)=P$. Prove that if $P\in C$ is an ...
0 votes
1 answer
62 views

Polar curve of a non singular projective cubic curve with respect to inflection point is union of two distict lines.

Hey guys I am currently struggling with a question that goes as follows. Let $C$ be a non-singular projective cubic and let $p \in C$ be an inflection with tangent line $T$. Show that the polar curve ...
-1 votes
0 answers
36 views

What is the Gergonne-Noether theorem? [closed]

I'm reading a text in which the author refers to the Gergonne-Noether theorem, but I'm not aware of this theorem. Could somebody please point me to a reference? Thanks
0 votes
0 answers
25 views

Finiteness of the intersection number [duplicate]

I am taking a course on Algebraic Curves following Gathmann and I am trying to solve exercise 2.7(b) which reads as follows: $F,G$ two curves with no common components through the origin, then every ...
2 votes
1 answer
80 views

Why does the MMP terminates when the canonical divisor is nef?

I have recently taken interest in Mori's Minimal Model Program (MMP) and I struggle to figure out why it stops when the canonical divisor $K_X$ of our variety $X$ is nef. For now, I have understood ...
-1 votes
1 answer
62 views

What is the canonical divisor of a genus 2 curve? [closed]

Let $C$ be a genus 2 smooth curve and let $K_C$ be a canonical divisor. I know by Riemann Roch theorem that $deg(K_C)=2g-2=2$. Can I specifically say what $K_C$ is linearly equivalent to? For example,...
1 vote
1 answer
54 views

Find isomorphism of elliptic curves in Weierstrass form

I have the following two elliptic curves over an algebraically closed field of characteristic distinct from 2: $$E:y^2=x^3+4x^2+2x\quad \quad E':y=x^3-8x^2+8x$$ I want to find an isomorphim $\psi:E'\...
0 votes
0 answers
31 views

Alternative way to compute degree of Frobenius endomorphism

Let $p$ be a prime, $q=p^r$, $K=\mathbb{F}_{q}$, $E/K$ an elliptic curve and let $\phi$ be the $q^{th}$ Frobenius endormorphism i.e. $\phi=(x^q,y^q,1)$. I want to show that $\phi$ has degree $q$. I am ...
1 vote
0 answers
476 views

Hartshorne IV 2.2

This is exercise 2.2 in Hartshorne's AG: What I have done: a): Apply Hurwitz theorem and observe that ramification index is no greater than degree. b): There is $ k[x]\rightarrow k[x,z]/(z^2-\prod (...
4 votes
1 answer
993 views

plane curve of degree 4

I was dealing with exercise IV.3.2 in Hartshorne, which says the following: Let $X$ be a plane curve of degree 4. a) Show that the canonical divisors on $X$ are exactly the hyperplane divisors. b) ...
3 votes
1 answer
133 views

Are these elliptic curves over $\mathbb{Q}(\sqrt{2})$ isogenous?

I have the following elliptic curves: $$E:y^2=x^3-\sqrt{2}x \quad \quad E':y^2=x^3-2x$$ over $\mathbb{Q}(\sqrt{2})$. I want to determine whether they are isogenous. I have a few strategies to do this ...
0 votes
0 answers
56 views

Degree of $[\rho]^2-[\rho]$ with $\rho^3=1$

I have the elliptic curve $E:y^2=x^3+B$ where $B\in K^\times$ and $K$ is a field of characteristic distinct than $2,3$. I have the map $\mu:=[\rho]^2-[\rho]$ and I want to compute its degree. My ...
3 votes
1 answer
44 views

Problem 1.49 (b): Algebraic Curves - Fulton

I'm trying to solve this exercise for quite some time, but I need help. Let $L$ be a field, $k$ an algebraic closed subfield of $L$. (a) Show that any element of $L$ that is algebraic over $k$ is ...
0 votes
0 answers
52 views

Genus of the curve $x^a + 1 = y^b$ is always positive

I believe the statement in the question is true whenever $a > b \geq 2$, does anyone know of a way to prove this? By homogenizing the equation we get $$F:X^a+Z^a = Y^bZ^{a-b}.$$ Then by computing $\...
1 vote
1 answer
235 views

J-invariants, Elliptic Curves, Cross Ratio [closed]

Let $E$ be an elliptic curve in $\mathbb P^2$ and $p$ be any point on $E$. From $p$ we can draw four tangent lines to $E$ and let $\lambda$ be the cross ratio of their slopes. How can we prove that $\...
3 votes
1 answer
66 views

Computing torsion subgroup of elliptic curve

Compute the torsion subgroup of the elliptic curve $y^2=x^3+5x^2+3x+7$. I am only used to computing torsion groups when our equation is in 'short Weirstrass form'; i.e. $y^2=x^3+Ax+B$ for integer $A,...
2 votes
0 answers
65 views

Universal property of Abelian-Jacobi Map/Jacobi variety for Riemann Surfaces

I have a question about universal property of Abel Jacobi Map and the Jacobi variety in the (classical) context of Riemann surfaces / complex smooth proper curves. Let $C$ be such RS/complex sm curve $...
0 votes
1 answer
30 views

Lemniscate of Bernoulli using Watt's linkage

The lemniscate of Bernoulli is a curve which can be defined as all points $P$ with $\overline{PA} \cdot \overline{PB}=2c$ with two given points $A$ and $B$ at distance 2c (see wikipedia). One way to ...
0 votes
1 answer
45 views

A problem about morphisms from a genus 2 curve to a quartic curve.

I am to answer the following question. Let $X$ be a projective non-singular curve of genus $2$, and let $K$ be an effective canonical divisor on it. Pick any two points $P_1,P_2$ on $X$ with $K \not \...
0 votes
1 answer
40 views

Find the equation of the closure of a curve in the projective space [closed]

Let k be an algebrically closed field and consider $C \subset \mathbb{A}^2(k)$, the curve of equation $f(X,Y)=0$ with $f(X,Y)=X(X-1)(X+1)-Y^2$. We want to find the equation of its closure in $\...
0 votes
1 answer
33 views

Question about proof regarding holomorphic maps between Riemann surfaces

So in our lecture notes, we have this proof; My question is, why the discussion on accumulation points? It seems to me like no part of this proof required any notion of accumulation points. Even the ...
6 votes
1 answer
202 views

Prove that $\int_0^1\frac{dx}{\sqrt{4x^5+1}}=\frac{\Gamma(\frac 3{10})\Gamma(\frac 65)}{2^{2/5}\Gamma(\frac 12)}\frac{5+\sqrt 5}{10}$

Using some mathematical software like mathematica, one can find that $$\int_0^1\frac{dx}{\sqrt{4x^5+1}}=\frac{3+\sqrt5}{2}\int_1^{+\infty}\frac{dx}{\sqrt{4x^5+1}}=0....
2 votes
2 answers
121 views

Find the area enclosed by the curve $(x^2+y^2-1)^3-x^2y^3=0$

The curve $$(x^2+y^2-1)^3-x^2y^3=0$$ forms a heart shaped curve. I want to find the area enclosed by it. This curve is a sixth degree algebraic curve, so y cannot be found and x cannot be found. The ...
1 vote
2 answers
83 views

Maximal torsion-free subgroup of an elliptic curve

Let $E$ be an elliptic curve over a field $k$. Then we have the subgroup $E_{\text{tors}}$ which denotes all the torsion elements of $E$. This is the biggest subgroup that only contains torsion ...
0 votes
0 answers
25 views

Pseudoeffectivness (or nefness) on a curve.

In a proof about varieties of higher dimension we do an induction, so we consider the case where the dimension of a variety is $n=1$. The thing is that we are supposing that some sivisor $D$ on our ...
5 votes
1 answer
70 views

Is the curve $y^7=x^2(x-1)^2$ hyperelliptic?

When playing around with genus $3$ curve $X$ which has an automorphism $\sigma$ of order $7$. Using the Riemann-Hurwitz formula, one find the map $X\to Y:=X/\langle\sigma\rangle$ is a degree $7$ map ...
0 votes
0 answers
18 views

Sequence involving plane curves is exact

I'm reading through Andreas Gathmann's Algebraic Curves and there is this affirmative on Proposition 2.10 at page 14: Let $P\in\mathbb{A}^2$, and $F, G, H$ curves (or polynomials). If $F$ and $G$ have ...
2 votes
1 answer
64 views

What's the distinction between an elliptic curve and its rational points?

I'm confused by this statement: "Be careful that you understand the distinction between the elliptic curve E and the group E(k) of its k-rational points. The group law is defined for the curve E, ...
3 votes
0 answers
16 views

Spiric sections by imaginary planes

To quote a previous question on spiric sections: The spiric section is the curve obtained by slicing a torus along a plane parallel to its axis. In comparison, Wikipedia uses an algebraic definition ...
2 votes
1 answer
40 views

Correspondence between basepoint free linear systems and Holomorphic maps to $\mathbb{P}^n$

I am currently reading's Miranda's book on Riemann Surfaces. I have a question about codimension 1 subspaces of $L(D)$. I am stuck on the proof of Propoisiton 4.15 in chapter 5. Questions regarding ...
1 vote
1 answer
100 views

Example for intersection, multiplicity and tangents for projective plane curves

We recently got introduced to projective plane curves in our class, however we just defined it and never really talked about examples. And when I am trying to come up with a just any curve, ...
4 votes
1 answer
103 views

What geometry is preserved by the translation maps on elliptic curves?

Let $E$ be an elliptic curve (over some field). For any $P \in E$, there is a translation map $T_P: E \to E$ given by $Q \mapsto P+Q$. This map is rational (i.e. the coordinates of $T_P(Q)$ are ...
1 vote
1 answer
34 views

Problem 6.8 in Fulton's Algebraic Curves: the nonvanishing locus of a regular function is open

Question: Let $U$ be an open subset of a variety $V$, $z\in k(V)$. Suppose $z\in O_p(V)$ for all $P\in U$. Show that $U_z = \lbrace P\in U \mid z(P) \ne 0 \rbrace $ is open. Attempt: Initially I want ...
0 votes
0 answers
37 views

Properties of field extensions of finite transcendence degree

Let $K/F$ be a field extension of transcendence degree $n\ge 2$. If $E$ is a subfield of $K$ such that $F\subset E \subset K$ and the transcendence degree of $E$ over $F$ equals $n-1$, then $K$ is a ...
1 vote
1 answer
95 views

Number of lines tangent to multiple points of a non-singular projective plane curve

Let $C:F(x,y,z)=0$ be a projective plane curve, where $F$ is a homogenous polynomial of degree $d$. I want to show that there exists $p\in \mathcal{P}_2$ such that all lines through $p$ are tangent to ...
16 votes
1 answer
947 views

Exercise 1.24 from Joe Harris' Algebraic Geometry: First Course

I have a question about solvability of Exercise 1.24 (p. 14) from Joe Harris' Algebraic Geometry: A First Course and correctness of following 'synthetic' construction which according to the book (or ...
0 votes
0 answers
44 views

Jacobian of plane curve

I am trying to prove the following: Let $C$ be a reduced singular curve in $\mathbb{P}^2$ such that the lines of its Jacobian matrix are linearly dependent. Then I would like to prove that $C$ is the ...
1 vote
0 answers
24 views

Does a kind of real plane algebraic curve always have a factorization corresponding to connected components?

$f(x,y)$ is a real polynomial such that in the equation $f(x,y)=0$ we can express $x$ in $y$ with composition of polynomial functions and square roots, and can express $y$ in $x$ with composition of ...
0 votes
2 answers
164 views

Rational Parameterization of Quartic Curve (Variety)

I am trying to find a rational parameterization of the curve (variety) $$x^4+a^2x^2=x^2y^2+(h^2+a^2)y^2$$ If I have done my math correctly, this curve has a singularity of multiplicity 2 at the origin,...
2 votes
0 answers
58 views

Compact Riemann surfaces as algebraic curves

I'm trying to grasp proofs of the fact that compact Riemann surfaces are algebraic curves. It seems as though no textbook fully treats this, and I wanted to ask about what was going on. On one hand, ...
2 votes
0 answers
26 views

totally ramified covering of $\mathbb{P}^1$ with 3 branch points.

I'm trying to construct a totally ramified covering (of order $d$) of the complex projective line $\mathbb{P}^1$ with exactly 3 branch points $0,1,\infty$. Here is what I'm trying: consider the smooth ...

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