# Tagged Questions

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 ...

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### 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 $H^1(\tilde{C},\pi^\ast\mathcal{O}_C)=0$...
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### Intersection of following pair of parabolas at infinity? [closed]

What is the intersection multiplicity of the following pair of parabolas at infinity:$$y = x^2,\text{ }y = x^2 + 1?$$
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### 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 ...
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### 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 ...
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### 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 $\bar{k}(x)$.
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### $\mathbb P^1\times \mathbb P^1$ 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 ...
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### 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 ...
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### 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 ...
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### 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 $\mathbb{P}^1$?
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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 ...
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### “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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\eta$. ...
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### 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 ...
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### 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) - ...
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### 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 ...
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### 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 \sum_{P\in \phi^{-1}(Q)}e_{\phi}(...
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### 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 ...
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### 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 ...
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### 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 $z=\frac{|w|^2-1}{|w|^2+1}$. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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}(\...