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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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 (...
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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?
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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 $E$ ...
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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
$$P_t(\gamma(s)) =0, \quad \forall (t,s)\in I\times ...
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Computing degree of $x$ map for elliptic curve given by Weierstrass equation
Suppose $E$ is an elliptic curve given by the Weierstrass equation
$$
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]...
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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,...
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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 '...
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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 ...
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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 :
$$(y(y+1)+1)^2+1=100x\...
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Number of bounded and unbounded components of nonsingular real algebraic curve $y^2 - p(x)$
I would like to know if someone can kindly verify my solution to Problem 22.2 from MIT's online course on geometry and topology in the plane. Let $C = \{(x,y) \in \mathbb{R}^2 : f(x,y) = 0\}$ be a ...
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If the number of intersection of two conics is an odd number, the quadratic forms are not simultaneous diagonalizable
I'm trying to do Exercise 3.6 at the end of this pdf (in $\Bbb CP^2$):
Show that the two quadratic forms
$$x^2+y^2-z^2, \quad x^2+y^2-y z$$
cannot be simultaneously diagonalized.
Attempt 1:
Their ...
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Polar curve of a non singular projective cubic curve with respect to inflection point is union of two distict lines.
Hey guys I am currently struggling with a question that goes as follows.
Let $C$ be a non-singular projective cubic and let $p \in C$ be an inflection with tangent line $T$. Show that the polar curve ...
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Finiteness of the intersection number [duplicate]
I am taking a course on Algebraic Curves following Gathmann and I am trying to solve exercise 2.7(b) which reads as follows:
$F,G$ two curves with no common components through the origin, then every ...
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What is the canonical divisor of a genus 2 curve? [closed]
Let $C$ be a genus 2 smooth curve and let $K_C$ be a canonical divisor. I know by Riemann Roch theorem that $deg(K_C)=2g-2=2$.
Can I specifically say what $K_C$ is linearly equivalent to? For example,...
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Find isomorphism of elliptic curves in Weierstrass form
I have the following two elliptic curves over an algebraically closed field of characteristic distinct from 2:
$$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'\...
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