# 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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### Understanding an exercise from Fulton's Book on Algebraic Curves

I am reading Fulton's book Algebraic Curves. Currently I am working on a specific problem (2.43), and I have doubts about my work and would appreciate another opinion(s). Assume $p$ is the origin ...
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### Why the image of quadratic Veronese map has the form $v\cdot v$?

It says that the image of quadratic Veronese map $v_2(P^1)$ is the subset of $P(Sym^2V)$ with the form $v\cdot v$. Isn't it has the form $x^2+xy+y^2$? So how can it be some $v\cdot v$?
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### Etale map from a variety to an elliptic curve

I read this sentence and I can't see why it is true. Let $E$ be an elliptic curve over an algebraically closed field $k$, $f\colon Y\to E$ an etale map; then $Y$ is also a curve over $k$. Can ...
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### Isomorphism of the affine circumference over certain fields.

Let us consider the coordinate ring of the circumference $$A:=K[X,Y]/(X^{2}+Y^{2}-1),$$ and let us suppose that $K$ is infinite but not necesarilly algebraically closed. I wonder if $A$ is ...
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### Ideal of the hyperbola in a field that is not algebraically closed.

Let $K$ a field not necessarily algebraically closed. I would like to find the coordinate ring of the hyperbola $$V(XY-1)\subseteq K^{2}.$$ If the field was algebraically closed we could use Hilbert'...
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### Definition of algebraic cusp

Is not it true that for planar curves, an $\textit{algebraic cusp}$, say at the origin, is the one that can be locally represented by $y^n=x^m$ with $m,n\in\mathbb N$ and $(m,n)=1$? What is a ...
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### Line through 2 flex passes through a third flex

(in $\mathbb P^2$)Show that a line through two flexes on a cubic passes through a third flex. I've tried to solve this problem using the corollaries of the Max Noether theorem (which talks about ...
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### A curve has the equation $y=x^3 - 4x^2 - 3x + 17$. What are the x-coordinates of the point(s) on this curve where the tangent is parallel to 4y=7x-11.

A curve has the equation $$y=x^3 - 4x^2 - 3x + 17$$. What are the x-coordinates of the point(s) on this curve where the tangent is... (a) parallel to $4y=7x-11$ (b) horizontal
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### Does anyone know of any good sources on the algebraic theory of abelian varieties?

I have a copy of Mumford's book, but as a final year undergraduate I am finding it to be a little too dense as a starting text. Something lighter would be appreciated to get an intuition before ...
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### Finding a cubic function with one real root given its graph.

When given a cubic graph with one real root. I need to find the equation of that graph using the function $$y=a(x-s)(x^2+bx+c)$$ where a, b,and c are unknowns. The y intercept is therefore $t = -asc$ ...
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### Linearly normal embedding and varietes lying on quadrics

Let $X\subset\Bbb{P}^N$ be a smooth algebraic variety and assume that $X$ is not contained in a hyperplane. The embedding $i\colon X\hookrightarrow\Bbb{P}^N$ is called linearly normal if the linear ...
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### Finding and analyizing the singularities in Affine and Projective space

Hi guys I am working this $F(x,y,z)=xy^4+yx^4+xz^4$ I need to find the singularities in affine and projective space and find the multiplicity of them.I would really appreciate some help tips. So ...
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### Proving a projective quadric is nonsingular

Let $K$ be an algebraically closed field of characteristic $\neq 2$. Let $C$ be an irreducible quadric curve in $\mathbb{P}^2$, i.e. $C = Z(F)$ where $F$ is an irreducible degree 2 form. I think we ...
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### Finding length of curve $y^2 = 64(x+3)^3$ for $0 \le x \le 3$

Not getting the right answer for this, can someone point me to where I'm going wrong?
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### Abstract regular curve over non-algebraically closed field

In Hartshorne chapter I.6 is discussed the construction of the abstract nonsingular curve as part of the proof for the well known correspondence between complete regular irreducible algrebaic curves ...