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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Blow up and hyperelliptic curves.

Let $C'$ be a non singular affine curve $y^2=x^5+3$ over $\mathbb{C}$. $C'^\#$ be its projective closure : $Y^2Z^3=X^5+3Z^5$. It has singular point at $\mathcal{O}'=(0:1:0)$. On the other hand, let $C$...
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Zeta functions of hyperelliptic curves

I have been wondering recently about the geometric information encoded in the zeta function of a (smooth, projective) variety over a finite field - or in its étale cohomology (i.e. l-adic cohomology) ...
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Automorphism group of the Klein quartic over field of characteristic 2,3,7(Hartshorne exercise IV.5.7)

Let $k$ be an algebraically closed field of characteristic $p$, and $X$ be the plane quartic curve defined by $$x^3y+y^3z+z^3x=0,$$ which is the so-called Klein quartic. Hartshorne claims in exercise ...
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equations for cuspidal curves

The space of morphisms $\mathbb P^1\to \mathbb P^2$ of degree $3$ is a Zariski open subset $U$ of $$ \mathbb P(H^0(\mathbb P^1, \mathcal O(3))^{\oplus 3}) \cong \mathbb P^{11} $$ the generic rational ...
Aitor Iribar Lopez's user avatar
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An exact sequence involving the Picard group.

I am studying ''The Arithmetic of Elliptic curves'' curves by Silverman and I am stuck in the problem 2.13: Let $K_0$ a field $K=\bar{K}_0$ its algebraic closure and $C$ a smooth projective curve ...
Marcos's user avatar
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Regular singular points on curves

The reference for this post is $11^{th}$ section of the article, N. Katz, Nilpotent connections and the monodromy theorem. Suppose $k$ is a field of characteristic $0$ and $K/k$ is the function field ...
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On well-definedness of Shimura curve.

Suppose that we have a quaternion algebra $D$ over a totally real number field $K$ such that $[K \colon {\Bbb Q}] = {\mathrm{odd}}$. We assume that $D$ splits everywhere at finite places of $K$ and at ...
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Is there a method to determine whether two algebraic curves are related by an euclidean isometry?

In differential geometry we have the fundamental theorem of curves for two curves given parametrically: If two plane curves, parametrized by their arc length, have the same curvature (as a function ...
roymend's user avatar
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Genus of $y^3=(x^2+1)^2(x^3-1)$

I'm trying to proving that the genus of the normalization of $C:y^3=(x^2+1)^2(x^3-1)$ is 4 by using the Riemann-Hurwitz formula on the projection map $\pi:C\rightarrow\mathbb{P}^1$ which sends $[x:y:z]...
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Blow up $\operatorname{Pic}^3(C)$ along $C$.

I want to solve this problem from E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris (auth.) - "Geometry of Algebraic Curves Volume I" I can show that the fibers of $\{(K_C-P) \in \...
Kirill Losev's user avatar
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When is relative Hilbert scheme $\text{Hilb}_{X/S}^{p(t)}$ flat over $S$?

Let $S$ be a scheme of finite type over the complex numbers $\mathbb{C}$ and let $X\subset S\times\mathbb{P}^r$ be a projective family over $S$. In the book "Geometry of Algebraic Curves Volume II" ...
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Incidence geometry with curves instead of lines

Let $K$ be an algebraically closed field. Say that a subset $C$ of the affine plane $K^2$ is an irreducible curve if there is an irreducible polynomial $f\in K[X,Y]$ (where $X$ and $Y$ are ...
Pierre-Yves Gaillard's user avatar
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Connected levels and polynomials submersions

Is it true that a polynomial submersion $ p: \mathbb{R}^2 \to \mathbb{R}$ of degree $n$ has at most $n$ connected components on each level? I think I have a proof, can someone point me out any ...
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Smooth quartic surface in $\mathbb{P}^{3}$ that contains a smooth curve of genus 2 and degree 6.

I am reading through an article by Matsumura and Monsky on Automorphisms of Hypersurfaces in which they state that there exist quartic surfaces is $\mathbb{P}^{3}$ which have infinite automorphism ...
user551642's user avatar
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Cardinality of a supersingular hyperelliptic curve of genus $2$ over $\mathbb{F}_q$ (q square)

I am trying to calculate the cardinality of a genus $g=2$ curve under certain hypotheses, but I do not know if the information I have is enough to infer this number or a bound for it (without using ...
toorandom's user avatar
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Is any regular algebraic curve in the plane union of closed curves?

This is related to the attempt I am doing trying to prove that any polynomial lemniscate is the union of regular closed curves. Recall by the way that a polynomial lemniscate is a curve of the form $\{...
Andrés Ibáñez Núñez's user avatar
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Affine plane curves classification

Define an affine plane conic as $\operatorname{Spec}A$ where $A=k[x,y]/(f)$ and $f$ a quadratic polynomial with no multiple factors. Define an equivalence relation on the set of alline plane conics ...
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Vector space of differential forms on a curve

Why is the space of differential forms on a curve (defined on the fraction field of the coordinate ring) a one-dimensional vector space over the function field of the curve? (I am uncomfortable with ...
Shravan Patankar's user avatar
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(first) cohomology $H^1$ of sheaves on curves

Say we have a reduced, irreducible, complete curve $X$ over a field $k$ and a sheaf $\mathcal F$ on $X$. I am interested in understanding the cohomology groups of $\mathcal F$. By flat base change, ...
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Cubic projective curves without rational points

I'm trying to study Elliptic Curves currently, and in the book I'm reading there seems to be one question they keep gravitating around but never actually even attempt to answer, even though it seems ...
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Deformations and blow-down

I would like to understand how blowing down affects (first order) deformations. For a concrete example, set $D=k[\epsilon]/(\epsilon^2)$ and consider in $\mathbb{A}^2_D=\operatorname{Spec} D[x,y]$ ...
Sonner's user avatar
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What do $L$-functions of curves over $\mathbb Q$ tell us about the curve

Following up this thread: $L$-function of an elliptic curve and isomorphism class I'd like to ask some more questions for the case of smooth projective curves $C$ over $\mathbb Q$ To be more precise,...
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structural (algebraic) sheaf on a Riemann surface "inside" the sheaf of holomorphic functions

This question consists of $3$ points, and each of them deals with the relationship between the "algebraic" structure on a projective curve and its "analytic" structure. I know that this argument has ...
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Group associated with this curve

Consider an Elliptic curve with a singularity (double point) over $\mathbb{F}_p$ $$y^2=x^2(x-a) $$ where $a$ is a quadratic residue in ${\mathbb{F}_p}^*$. We can easily count the number of (non-...
xyz's user avatar
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Order of point in divisor

Let $y^2=f(x)$ be hyperelliptic curve over $k$ and $(a,\sqrt{f(a)})$ point on the curve and let $b\in k$. I would like to prove that divisor of the function $g(x,y)=\frac{x-b}{x-a}$ is equal to $$...
Meow's user avatar
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Proof of Chapter 2 Proposition 2.6a in Silverman Arithmetic of Elliptic Curves

The following is Proposition 2.6(a) in Silverman's AEC: "Let $\phi: C_1 \rightarrow C_2$ be a nonconstant rational map of smooth (projective) curves over an algebraically closed field $K$. Then $\...
spadey's user avatar
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Does roots of L-polynomial of curve being roots of unity imply the curve is supersingular?

If $C$ is a supersingular curve over $\mathbb F_q$, then $\frac1{\sqrt q}$ of roots of $L$ function are roots of unity. What about converse? If $\frac{1}{\sqrt q}$ of roots of $L$ polynomial are ...
vudu vucu's user avatar
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Explicit computation of Serre duality

Given a projective non-singular curve $X$, the a Serre duality asserts an isomorphism between $H^0(X,\Omega^1_X)$ and dual of $H^1(X,\mathcal{O}_X)$. My question is how to compute the dual elements in ...
Darius Math's user avatar
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Linking regularity of ideal sheaf with Fitting ideals sheaf

I'm reading Eisenbud's book The geometry of syzygies and I'm quite struck undestanding the argument proposed in Chapter 5, in the section named "Fitting ideals". Remember that a coherent sheaf $\...
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What is the intuition behind the statement "A rational map from a smooth curve can be extended to a regular one"?

I know how to prove this using a local equation for the non-regularity locus and the that the local ring at a smooth point is a UFD, and today I saw an example of this happening in practice: Let $V : ...
Elle Najt's user avatar
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Use of the Bezout's theorem in Abstract Algebra

The Bezout's theorem: Let $C$ and $D$ be two plane curves described by equations $f(X,Y) = 0$ and $g(X,Y) = 0$, where $f$ and $g$ are nonzero polynomials of degree $m$ and $n$, respectively. ...
Mi-lee Wilson's user avatar
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Do retractions exist only on rational curves?

I read the following in Eisenbud's Commutative Algebra with a view .... Let $k$ be an algebraically closed field. Recall that a retract is a morphism which is a retraction of the inclusion, and $X$ is ...
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Question about cusp cubic example in Hartshorne

In Hartshorne's Algebraic Geometry, in Chapter II.6 on Divisors he computes the Cartier class group (denoted $\operatorname{CaCl}$) of the cuspidal cubic cut out by $y^2z=x^3$ in $\mathbb{P}^2$. He ...
Atticus Christensen's user avatar
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Calculate the singular points of affine curve

I want to calculate the singular points of the affine curve $$f(X,Y)=(1+X^2)^2-XY^2 \in \mathbb{C}[X,Y]$$ The point $P=(x,y)$ is singular $\Leftrightarrow$ If $x=0$ we find $y=0$ and then from the ...
evinda's user avatar
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Pushing forward vector bundles on a plane curve via projection from a point

Let $C \subset \mathbb{P}^2$ be a smooth plane curve, $P \in \mathbb{P}^2$ is point not on $C$, consider projection from this point $$ \pi :\mathbb{P}^2 - \{P\} \to \mathbb{P}^1, $$ and restrict this ...
Alex's user avatar
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Where do divisors on curves come from?

I've been learning about Riemann surfaces and the Riemann-Roch theorem, and I've been able to experience some of its great power. However, I'm curious as to what the history behind divisors on curves ...
Supersingularity's user avatar
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When do two integral superellipses have 'nice' intersections?

A recent question posed the nonlinear system \begin{cases} 3x^3+4y^3=7\\ 4x^4+3y^4=16 \end{cases} for real $(x,y)$ and asked for the sum $x+y$. As noted by commentary in the question, this regrettably ...
Semiclassical's user avatar
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Simple Branched covering over sphere.

A simple branched covering is a branched covering with branching points of degree at most 2, in some context, it is also required to have at most one branching point in each fiber. My question is ...
ZZY's user avatar
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Inequality involving multiplicities of points introduced via Quadratic Transformations of a Plane Curve

I've been learning about the resolution of singularities for plane curves, and have become stuck at exercise 7.15 of Fulton's Algebraic Curves (page 91 of the PDF). The question is: Let $F=F_1, \...
Craig's user avatar
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Lifting extensions of sheaves

Consider a curve $X_0$ over an algebraically closed field $k$. Let $A$ be a local Artinian ring over $k$ and $X$ a scheme over $A$ with $X \otimes_A k = X_0$. For a sheaf $F$ on $X$ we denote by $F_0$ ...
boxdot's user avatar
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Vanishing of sections and special divisors

Let $L$ be a line bundle on a smooth complex projective curve $X$. Suppose we have vector subspaces $$U\subset V\subset H^0(X,L),\,\,\,\textrm{and}\,\,\,\dim\, U\leq k,\,\,\dim\,V=k+1.$$ I wonder if ...
Brenin's user avatar
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Group actions on Čech cohomology

Suppose we have a curve $X$ and a group $G$ acting on $X$. Then one has an induced action of $G$ on the sheaf cohomology of $\mathcal O_X$. I wondered what one can say about the group action on the ...
Joe Tait's user avatar
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Are there Galois covers of curves branched at 1 point?

Let $G$ be a finite group, not necessarily abelian. Is there any smooth algebraic curve $C$, with an action of $G$ on $C$, such that the natural quotient map $C \to C/G$ is branched at precisely one ...
calc's user avatar
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When is an intersection of varieties finite

Consider the general Bezout's theorem: If $p_1 \ldots p_n$ are polynomials with degrees $d_1,\ldots, d_n$ in $\mathbb{R}[x_1,\ldots,x_n]$, with $V = \{a=(a_1,\ldots, a_n) | p_i(a) = 0, \forall i\}$ ...
user77375's user avatar
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If $X\times C$ is isomorphic to $Y\times C$, does it follow that $X$ is isomorphic to $Y$

Let $k$ be a field. Let $X$, $Y$ and $C$ be varieties over $k$ such that $X\times_k C $ is isomorphic to $Y\times_k C$. Assume that $C$ is a curve. (The varieties $X$, $Y$ and $C$ are assumed to be ...
Harry's user avatar
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Find an explicit isomorphism from a curve of genre zero to the Riemann sphere

I can't figure out this exercise: i have this singular curve in $\mathbb{P}^2\mathbb{C}$ given by $\{[X,Y,Z]\in \mathbb{P}^2\mathbb{C}:X^2Y^2+Y^2Z^2+X^2Z^2=0\}$, I have shown that its ...
tigu's user avatar
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Degree-$p$ étale covers of affine line

Let $k$ be an algebraically closed field of positive characteristic $p$. Suppose that $F$ is a monic irreducible polynomial over $k[x]$ of degree $p$ whose discriminant is an element of $k$. Is then $...
mainomai's user avatar
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Reference request: The ring $k[x_1, \dotsc, x_n]/I$ as a limit of functions with distinct zeroes

p. 152-153 of "Using Algebraic Geometry" (2nd. ed) by Cox, Little, and O'Shea says: [S]uppose, that a collection of $n$ polynomials $f_1, \dotsc, f_n$ has a single zero in $k^n$, which we ...
Fred Akalin's user avatar
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122 views

Checking that a differential form is holomorphic on an algebraic curve and that it extends to a holomorphic form on its compactification

Consider the following algebraic curve: $$C=\left\{ (x,y) \in \mathbb{C}^2 | x^3+y^3+3\lambda xy + 1=0 \right\} $$ …where $\lambda^3 \neq -1$. I think this is called the Hesse pencil, and it’s a ...
Gokimo's user avatar
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On Siegel's Theorem in integer points.

Siegel’s theorem on integer points: Let $P \in {\mathbb{Q}}[x,y]$ be an irreducible polynomial of two variables, such that the affine plane curve $C = \{ (x,y): P(x,y)=0\}$ either has genus at least ...
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