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 ...

learn more… | top users | synonyms

1
vote
1answer
77 views

Cohomology and normalization of a curve

Suppose $C$ is a projective curve such that $H^1(C,\mathcal{O}_C)=0$ and let $\pi:\tilde{C}\longrightarrow C$ be the normalization morphism. How can I show that $H^1(\tilde{C},\pi^\ast\mathcal{O}_C)=0$...
3
votes
1answer
59 views

Intersection of following pair of parabolas at infinity? [closed]

What is the intersection multiplicity of the following pair of parabolas at infinity:$$y = x^2,\text{ }y = x^2 + 1?$$
0
votes
1answer
38 views

Maps induced by divisors and degree

I feel very stupid to ask this question but I've serached everywhere and I've not found a clear (to me) answer. A well known result in Curve theory (over $k=\bar{k}$) states that a divisor $D$ on a ...
1
vote
2answers
59 views

The Hirzebruch surface $\mathbb F_n$ contains a unique irreducible curve with negative self-intersection.

Consider the $n$-th Hirzebruch surface $\mathbb F_n:=\mathbb P_{\mathbb P^1}(\mathcal O_{\mathbb P^1}\otimes\mathcal O_{\mathbb P^1} (n))$, and let $C_0\subseteq \mathbb F_n$ a section of the ruling ...
0
votes
0answers
7 views

C: a smooth projective curve/k. $x\in\bar{k}(C).dx$ is a basis for $\Omega_C,\Rightarrow\bar{k}(C)$ is a finite separable extension of $\bar{k}(x)$.

Let C be a smooth projective curve over a field $k$ and $x\in\bar{k}(C)$. If $dx$ is a basis for $\Omega_C$, then $\bar{k}(C)$ is a finite separable extension of $\bar{k}(x)$.
1
vote
1answer
55 views

$\mathbb P^1\times \mathbb P^1$ is minimal

I'm trying to prove that the complex surface $\mathbb P^1\times \mathbb P^1$ is minimal. I'd like to prove it directly, namely by showing that there are not exceptional curves. Suppose that $E$ is a ...
5
votes
1answer
71 views

Degree of $f:\mathbb{P}^1_k\rightarrow \mathbb{P}^1_k$

Let $k$ be an algebrically closed field and consider $\mathbb{P}^1_k$ the $1$-dimensional projective space over $k$. My question is the following: Let consider $f:\mathbb{P}^1_k\rightarrow \...
2
votes
1answer
68 views

Geometrically ruled surface, sections and intersection numbers.

Consider a geometrically ruled surface $\pi: S\longrightarrow C$ where both $S$ and $C$ are projective, non-singular and complex ($C$ is obviously a curve). By using Tsen theorem one can show that ...
3
votes
0answers
52 views

Tricks to find $\dim H^0(O(D)(n))$

Let $X$ be a curve of genus $g\neq 0$ and $D$ a divisor on $X$. If $n \in \mathbb{Z}$ and $O(D)(n)=O(D)\otimes O(n)$ is the twist of $O(D)$ by $n$, there is a manageable proceedings to find the ...
3
votes
2answers
167 views

Understanding the connection between the projective space and the affine plane

Suppose we have a point $P=[x,y,z]\in \mathbb P^2$. Then at least one of the coordinates is not zero. Suppose $z\neq 0$. So we have write $P$ as $[x/z,y/z,1]$ and this point belongs to $(x/z,y/z)$ ...
1
vote
1answer
50 views

Looking for $\dim _{K}(m_{(0,0,0)}/m_{(0,0,0)}^{2})$ for certain algebraic variety.

Let $X=V(X_{2}^{2}-X_{0}^{2}X_{1},X_{1}^{3}-X_{0}^{4},X_{0}^{3}-X_{1}X_{2},X_{1}^{2}-X_{0}X_{2})\subseteq\mathbb{A}^{3}_{K}$. We denote $$ m_{(0,0,0)}=\{\overline{f}\in K[X]:f(0,0,0)=0\}, $$ where $K[...
2
votes
1answer
89 views

Logistic curve through three points

I need to find a logistic curve that passes through three points exactly. This means I cannot do a best fit but rather must use simultaneous equations. Essentially this is used to model population ...
1
vote
0answers
18 views

Is a multiple of a hyperelliptic curve hyperelliptic?

Let $C$ be a curve of genus 2 over $\mathbb{C}$. So $C$ is hyperelliptic, that is it admits a degree 2 map to $\mathbb{P}^1$. Is a power of $C$ say $nC$ hyperelliptic too, $n\geq 2$? It is not of ...
0
votes
0answers
12 views

How to prove $l(D)-l(-2(\infty)-D)=\text{deg}D-1$ without using Riemann-Roch theorem?

How to prove $l(D)-l(-2(\infty)-D)=\text{deg}D-1$ without using Riemann-Roch theorem, where $D$ is a divisor on $\mathbb{P}^1$?
4
votes
0answers
70 views

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 ...
2
votes
1answer
87 views

“path-connectedness” of an algebraic variety

Let $X$ be an irreducible algebraic variety over a field (supposed to be algebraically closed if necessary). How to proove that any two closed points of $X$ can be connected by a finite number of ...
0
votes
0answers
31 views

Establishing Linear Equivalence of Divisors on Curves

I am trying to do some questions from Geometry of Algebraic Curves by Arbarello-Cornalba-Griffiths-Harris. Here are some of the examples: Exercise A3: Curve: $y^2=x^3+1$. Let $\Gamma=C$ be the ...
3
votes
1answer
102 views

Looking for an affine curve not isomorphic to an affine plane curve.

I want to find an affine curve not isomorphic to an affine plane curve (as simple as possible). I am trying to find an affine curve $X\subseteq\mathbb{A}^{n}_{k}$ such that its coordinate ring is not ...
0
votes
0answers
73 views

Stalks of invertible sheaves on curves

I have just found out that I have maybe not understood very well what an invertible sheaf looks like. Let $X$ be a (regular, integral, separated, whatever you want) curve and $\mathcal{L}$ an ...
6
votes
2answers
151 views

About a theorem in Beauville's book.

Look at the following theorem (due to Castelnuovo) taken from Beauville's book on complex surfaces: I have a question about the behavior of $f$ on the exceptional curves generated by $\eta$. ...
0
votes
1answer
92 views

Question on the proof of Hartshorne IV.3.7

Proposition 3.7 in Hartshorne's Algebraic Geometry is the following. Let $X$ be a curve in $\mathbb P^3$, let $O$ be a point not on $X$, and let $\phi: X \rightarrow \mathbb P^2$ be the morphism ...
0
votes
0answers
17 views

Topology of the cuspoidal cubic

Let $C$ denote the set of solutions to $zy^2 = x^3$ inside of $\mathbb{C}P^2$. Someone told me that this space is homeomorphic to the pinched torus (or pinched sphere depending on how you pinch) - ...
0
votes
0answers
30 views

Morphism of ringed topological spaces

Let $f:X\rightarrow Y$ a morphism of ringed topological spaces. Let $V$ be an open subset of $Y$ containing $f(X)$. I want to show that there exists a unique morphism $g:X\rightarrow V$ whose ...
1
vote
0answers
20 views

The relationship between ramification index and “degree of maps between algebraic curves”

This problem is Exercise II.2.3. in Silverman's 'Arithmetic of elliptic curves' . Consider in the field $\mathbb{C}$. We need to verify directly that \begin{equation} \sum_{P\in \phi^{-1}(Q)}e_{\phi}(...
1
vote
1answer
32 views

How can I figure out what the log function being used based off the X and Y values?

I have a chart where Microsoft .NET has automatically scaled it using a (supposedly) log10 function of some kind. I need to figure out what formula they're using for the value at each tick mark. The ...
0
votes
1answer
14 views

Do problem weights change as the overall grade of an assignment is curved?

When I get papers back for class, there's often a question or two that I know I could make a case for getting credit for my answer, but ultimately decide it's not worth the extra % in the grand scheme ...
0
votes
1answer
30 views

Is this parametric equation describe a circle?

Let $w=\varepsilon\beta(t)-i\sqrt{\beta(t)^2-1}$, where $\beta(t)=\cosh t$ and $\varepsilon >0$. the parametric function is defined as $x+iy=\frac{2w}{|w|^2+1}$ and $z=\frac{|w|^2-1}{|w|^2+1}$. ...
2
votes
1answer
57 views

Example of two subvarieties of $\mathbb{P}^2$ that are isomorphic but not projectively equivalent.

Two curves $C_1$, $C_2 \subset \mathbb{P}^2$ are called projectively equivalent if there is a projective change of coordinates $\phi: \mathbb{P}^2 \to \mathbb{P}^2$ so that $\phi(C_1) = C_2$. What is ...
1
vote
1answer
56 views

Relation between curves on a surface and divisors

Given a projective surface $S$, the irreducible curves contained in $S$ are exactly, by definition, the prime (Weil) divisors of $S$. I was wondering what are reducible curves on $S$ in terms of ...
2
votes
2answers
88 views

Corollary of Riemann Roch theorem

I'm studying about Riemann Roch theorem for curves and its consequences (in algebaic gemoetry). In Hartshorne I've found an exercise equivalent to this assert: Let $D$ be an effective divisor of ...
0
votes
1answer
20 views

Any higher level maths or theories for epicycloids and/or hypocycloids?

For my 12 grade folio task on cycloids, I need to research hypocycloids and/or epicycloids. I need to consider: - exploring how the relative radii of the circles relate to the path - develop ...
1
vote
0answers
46 views

Degree of pull-back of locally free sheaves under normalization

Let $X$ be a projective, nodal curve, $\pi:\tilde{X} \to X$ be its normalization and $\mathcal{L}$ be an invertible sheaf on $X$. The question is: Is $\deg(\pi^*\mathcal{L})=\deg(\mathcal{L})$? As ...
3
votes
0answers
47 views

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 $\...
0
votes
1answer
33 views

If for any external point, exactly two tangents can be drawn to an algebraic curve, must the curve be a conic?

Yesterday, my teacher, while proving Poncelet's theorem, seemed to use the fact that if from any external point (external meaning, I assume $f(x,y)>0$ where $f$ is the polynomial of two variables ...
1
vote
1answer
58 views

Can a polynomial of degree 2 vanish on three different lines?

Suppose $p(x,y,z)$ is a homogeneous polynomial of degree 2. My question is: Can I have three distinct lines $L_1,L_2,L_3$ in the projective space $\mathbb{P}^2$ such that $p$ vanishes on every ...
0
votes
0answers
34 views

Self Intersection Formula

Let $\pi$: $X\to \Delta$ be a good degeneration of surfaces with degenerate fiber $X_0 = V_1 + ... + V_n$. Let $C$ be a component of the double curve $V_1 \cap V_2$ I am trying to understand why the ...
1
vote
0answers
48 views

Dimension of homogeneous polynomials passing through 4 points

Can anyone help me solve the following exercise? Let $p_1,p_2,p_3,p_4$ be distinct points in the projective plane $P^2$. What is the dimension of the vector space of homogeneous polynomials $f(x_0,...
2
votes
0answers
59 views

bases of a function field

I read this example Example 3.33: Consider the Hermitian curve with equation $x^{r+1}=y^r+y$ over the field $\mathbb{F}_{r^2}$. Here $Q = (0:1:0)$ and $X = x/z$, $Y=y/z$ is a monomial generating ...
1
vote
1answer
63 views

There exists a divisor linearly equivalent to $P$ not containing $P$

Let $X$ be a non-singular projective curve over $\mathbb C$ and let $P\in X$ be a closed point. Which is, in your opinion, the easiest way to prove that there exists a divisor $D$ of $X$ satisfying ...
1
vote
1answer
93 views

Every point lies on a unique secant through $C$

Let $C \subset \mathbb{P}^3$ be the twisted cubic (i.e., $C=\{(X_0^3:X_0^2 X_1:X_0 X_1^2:X_1 ^3) : (X_0,X_1) \in \mathbb{P}^1\}$). I need to show that every point $Q \in \mathbb{P}^3 \setminus C$ lies ...
1
vote
1answer
66 views

Generic point and pull back

Let's work over $\mathbb C$, Consider a finite morphism between two integral curves $f: X\rightarrow Y$, let $\eta_Y$ (resp $\eta_X$) be the generic points Questions: What is $X_{\eta_Y}=f^{-1}(\...
0
votes
0answers
15 views

Can the Klein Quartic be parameterized by meromorphic upper-half plane functions?

It is known that Elliptic Curves in canonical form can be parameterized by the Weierstrass elliptic function and its derivative on a suitably chosen lattice: $$[\wp'(z)]^2 = 4[\wp(z)]^3-g_2\wp(z)-g_3,...
2
votes
0answers
35 views

Basepoint free line bundles in $Pic^d(C)$ is an open set?

Let $C$ be a smooth projective curve over $\mathbb{C}$. Let $Pic^d(C)$ denote line bundles on $C$ with degree $d$. Is the subset of basepoint free line bundles an open set of $Pic^d(C)$? Many thanks!
1
vote
1answer
52 views

Simple question from Kunz's book Introduction to Plane Algebraic Curves

I'd like to ask a question that arose when I was reading the book Introduction to Plane Algebraic Curves by Kunz. In the following passage, how can we conclude that ker$\alpha = (q)$? I can see that ...
0
votes
1answer
52 views

Flat families and section of family of curves

Let $X$ be a smooth projective surface. Let $L$ be an ample line bundle on $X$. Consider $Y=\{(x,C) : C\in |L|, x\in C\}\subset X\times |L|$. Then $p_2:Y\longrightarrow |L|$ is a family of curves over ...
0
votes
0answers
16 views

Compute $\ell(W+P)$ and $\ell(W-P)$ for a place $P\in\mathbb{P}(F/K)$ of degree $1$ and any $W$ canonical divisor.

Let $F/K$ be a function field of genus $g$. Let $P\in\mathbb{P}(F/K)$ be a place of degree $1$ and $W$ be any canonical divisor. Determine $\ell(W+P)$ and $\ell(W-P)$. If $\ell(P)=\deg(P)+1$, then we ...
0
votes
0answers
34 views

a coordinate change of a quadric in P^2 to get one of 3 curves

I need to show that for a quadric curve in P^2(k) of degree 2,if the characteristic of k is not 2 ,then up to a coordinate change any quadric is one of the following :1) non-singular quadric 2) a pair ...
0
votes
0answers
35 views

Hurwitz bound is not sharp for curves of genus 4.

I'm trying to prove that for any curve $C$ over complex numbers $\mathbb C$ of genus 4 the number of automorphisms is strickly less then $252$. I proved this inequality for hyperelliptic ones. As for ...
0
votes
1answer
76 views

The translation map between elliptic curves is a rational map

I want to see a reference or a prove that the following map is a rational map: Let E be an elliptic curve,$P\in E$ and $T_p$ defined as $T_p:E\rightarrow E,\text{ }T_P(Q)=P+Q$. It is important ...
0
votes
1answer
18 views

Reverse-engineering a parametrization

Let's say you have a polynomial depending on complex parameters $A,B$: $$p(x,y) = (y + Ax)(y + Bx) + (A-B)^2$$ One parametrization of zero points of this polynomial is given by $$ x(t) = -(t + t^{-1})...