Skip to main content

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.

Filter by
Sorted by
Tagged with
1 vote
1 answer
54 views

Doubt in a proposition from Fulton's Algebraic Curves, section 5.5 (some criteria for Noether's condition)

I was reading section on Max Noether’s Fundamental Theorem in Fulton's Algebraic Curves and came across the following proposition which gives some sufficient criteria for Noether's condition to hold (...
Ajin Shaji Jose's user avatar
1 vote
0 answers
13 views

Does maximal contact hypersurface (m.c.h.) exist for any curve over a field of arbitrary characteristic, and if so, for what definition of m.c.h.?

Does a maximal contact hypersurface always exist for any curve over a field of arbitrary characteristic, and if so, for what definition of maximal contact hypersurface?
mathemusician's user avatar
1 vote
0 answers
29 views

Order of Picard groups of non-hyperelliptic algebraic curves

Let $q$ be prime. When $E/\overline{\mathbb{F}_q}$ is an elliptic curve, it is well-known that the group of $\mathbb{F}_q$-points of $E$ is isomorphic to the Picard group of degree $0$ divisors on $E$ ...
Tejas Rao's user avatar
  • 1,950
1 vote
1 answer
64 views
+50

References and useful results on continuous one-parameter intersection of algebraic surfaces

Consider a one-parameter family of polynomials $\{P_t\in \mathbb{R}[X,Y]\}_{t\in I}$ and a continuous curve $\gamma:J\to \mathbb{R}^2$. Suppose that $$P_t(\gamma(s)) =0, \quad \forall (t,s)\in I\times ...
Derso's user avatar
  • 2,773
1 vote
0 answers
48 views

Computing degree of $x$ map for elliptic curve given by Weierstrass equation

Suppose $E$ is an elliptic curve given by the Weierstrass equation $$ y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6 $$ I want to calculate the degree of the map $$ \varphi\colon E\to\mathbb{P}^1\qquad\quad[x,y,1]...
Navid's user avatar
  • 147
0 votes
1 answer
64 views

Smoothing projective nodal curve, is the general fiber smooth?

Proposition 29.9 of Hartshorne's Deformation theory states the following: A reduced curve Y in $\mathbb{P}^n$ with locally smoothable singularities and $H^1(Y,O_Y(1)) = 0$ is smoothable. In particular,...
maxo's user avatar
  • 53
0 votes
1 answer
82 views

What is a curve at $y=\infty$ mean?

On the wikipedia for the trident curve, $xy+ax^3+bx^2+cx=d$, two graphs are shown: Both are for the case where $a=b=c=d=1$, with the first matching what I find in desmos, but the latter being the '...
Eli Bartlett's user avatar
  • 1,685
1 vote
0 answers
31 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 ...
Framate's user avatar
  • 753
-2 votes
1 answer
65 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\...
Ranger-of-trente-deux-glands's user avatar
0 votes
0 answers
31 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 ...
Menander I's user avatar
1 vote
0 answers
36 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 ...
hbghlyj's user avatar
  • 3,047
0 votes
1 answer
69 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 ...
Dorelanië's user avatar
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 ...
Fernando Rabanillo Novoa's user avatar
-1 votes
1 answer
73 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,...
Angry_Math_Person's user avatar
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'\...
kubo's user avatar
  • 2,067
0 votes
0 answers
32 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 ...
kubo's user avatar
  • 2,067
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 ...
kubo's user avatar
  • 2,067
3 votes
1 answer
139 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 ...
kubo's user avatar
  • 2,067
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 ...
Gabriela Prampolim's user avatar
0 votes
0 answers
54 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 $\...
oleout's user avatar
  • 1,190
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,...
alidixon222's user avatar
2 votes
0 answers
67 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 $...
user267839's user avatar
  • 7,541
0 votes
1 answer
31 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 ...
garondal's user avatar
  • 889
2 votes
1 answer
95 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 ...
user1319604's user avatar
0 votes
1 answer
50 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 \...
John Robertson's user avatar
0 votes
1 answer
44 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 $\...
BillyJohny's user avatar
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 ...
B Kosta's user avatar
  • 146
2 votes
2 answers
126 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 ...
Matheman242's user avatar
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 ...
kubo's user avatar
  • 2,067
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 ...
raisinsec's user avatar
  • 463
5 votes
1 answer
74 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 ...
cybcat's user avatar
  • 786
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 ...
Mand's user avatar
  • 303
2 votes
1 answer
66 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, ...
popstack's user avatar
  • 291
3 votes
0 answers
17 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 ...
hbghlyj's user avatar
  • 3,047
4 votes
1 answer
104 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 ...
popstack's user avatar
  • 291
2 votes
1 answer
44 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 ...
pryvarick's user avatar
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 ...
leo's user avatar
  • 49
0 votes
0 answers
39 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 ...
merev's user avatar
  • 1
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 ...
User43029's user avatar
  • 1,264
1 vote
1 answer
99 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 ...
neilps2000's user avatar
2 votes
0 answers
63 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, ...
Ari Krishna's user avatar
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 ...
S.Gau at Math's user avatar
0 votes
1 answer
24 views

Relations of the rational points between two birational curves.

Let $X$ and $X'$ be two birational algebraic curves over $\mathbb{Q}$. Is it true (or not) that $X(\mathbb{Q})$ is Zariski dense iif $X'(\mathbb{Q})$ is Zariski dense ? If no, where can I find a ...
Marsault Chabat's user avatar
0 votes
0 answers
53 views

Smooth space filling curve over algebraically closed field

Let $X$ be a smooth projective geometrically integral variety of dimension $m \geq 1$ over $\mathbb{F}_q$ and a finite subset of points $F \subseteq X$. Then there is a way to construct a smooth ...
jb1403's user avatar
  • 23
0 votes
1 answer
97 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 ...
oti nane's user avatar
0 votes
0 answers
19 views

Embedding into generalized Jacobian

Let $k$ be an algebraically closed field and $X$ a smooth, projective curve over $k$. Consider a finite set $S$ of closed points of $X$ and let $U = X \setminus S$. Consider the divisor $\mathfrak{m} =...
Erich's user avatar
  • 245
0 votes
1 answer
60 views

If $\phi:V_1\rightarrow V_2$ is a morphism of varieties then $V_1\cong \phi(V_1)$

I am reading Silverman's The Arithmetic of Elliptic Curves. I am wondering if with the definition of morphism he gives, we can conclude that if $\phi:V_1\rightarrow V_2$ is a morphism of projective ...
kubo's user avatar
  • 2,067
2 votes
0 answers
36 views

Torsion group over $\mathbb{Q}$ of $Y^2=X^3+DX$ for $D\equiv 2\pmod{3}$.

Let $D\in\mathbb{N}$ with $D\equiv 2\pmod{3}$. Describe the torsion group over $\mathbb{Q}$ of the elliptic curve \begin{equation}Y^2=X^3+DX.\end{equation} Idea: Firstly we recall that for an ...
user avatar
0 votes
0 answers
57 views

Definition of algebraic curves

I'm doing a module on algebraic curves which follows Fulton's book and I'm very confused. Let K be a field. Right at the beginning of chapter 3 he defines an affine plane curve to be an equivalence ...
Moron3000's user avatar
2 votes
0 answers
119 views

Divisor of a differential form

Let $k$ such that $\operatorname{char}(k)=0$ and $C\subset \mathbb{P}^2$ be the smooth projective curve defined by $$X^4+Y^4+Z^4=0$$ I want to compute the divisor of $d\left(\frac{X}{Z}\right)$. My ...
kubo's user avatar
  • 2,067

1
2 3 4 5
51