# Tagged Questions

The study of geometric objects defined by polynomial equations, as well as their generalizations: algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc. Problems under this tag typically involve techniques of abstract algebra or complex-analytic methods. ...

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### When is diagonal morphism the diagonal map

Suppose $X, Y$ are schemes and $f : X \rightarrow Y$ a morphism and $\pi_1, \pi_2 : X \times_Y X \rightarrow X$ be the two projections and $\Delta : X \rightarrow X \times_Y X$ is diagonal morphism. ...
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### Varieties are isomorphism

Let $f:Y\to X$ be a birational morphism, Y is projective. Let $H$ be a very general ample divisor on $Y$. If $f^{*}f_{*}H=H$, is it true that $Y$ is isomorphic to $X$?
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### What is an “algèbre augmentée sur un corps?” (EGA I)

In EGA I, Chapter 0, (1.1.10), Grothendieck is giving examples of terminal objects in different categories. He says "dans la catégorie des algèbres augmentées sur un corps $K$ (où les morphismes sont ...
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### Do two rational parametric curves intersect only finitely many times?

Suppose there are two rational parametric curves $f = (f_1, \ldots, f_n)$ and $g = (g_1, \ldots, g_n)$ in $\mathbb{R}^n$. I read somewhere that such a parametric expression can always be transformed ...
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### Field of definition of a subvariety

Let $X$ be a variety over a field $\overline{k}$ (where $k$ is a field which is not algebraically closed). Each closed point of $X$ is defined over some finite extension $\ell/ k$, which means that it ...
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### Non-smooth curve in $\mathbb{A}^2$

In one of my exercises on Algebraic Geometry, I showed that the curve $X \subset \mathbb{A}^2$ defined by $x^3-y^2$ is irreducible but not smooth. Furthermore, they ask the following question that I ...
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### Dominant morphism on affine varieties

Let $X,Y\in \mathbb{A}^{n}_{k}$ affine varieties, I know that a morphism $f:X\rightarrow Y$ is dominant iff the correspondent morphism $\phi:k[Y]\rightarrow k[Y]$ is injective. How can I show from ...
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### Why is the index $i(\mathcal{L})$ of an ample line bundle on an abelian variety equal to $0$?

I've seen that here https://www.math.uchicago.edu/~ngo/Shimura.pdf there's a theorem called Mumford's Vanishing Theorem (Theorem 2.2.2) which says: Let $\mathcal{L}$ be a line bundle on $X$ (...
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### Parametric Equation of Elliptical Cycloidal Sine Curve

I am trying to find the parametric equations of a cycloidal curve, which, instead of using a circle, uses an ellipse to oscillate around a base circle. Below are equations of the standard, circular ...
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### Define hyperplane out of 4 points

Given the Cartesian co-ordinates (x,y,z,w) of 4 non-coplanar points: $P1:(x1,y1,z1,w1)$ $P2:(x2,y2,z2,w2)$ $P3:(x3,y3,z3,w3)$ $P4:(x4,y4,z4,w4)$ I want to find the equation of the hyperplane on which ...
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### Recommended books on commutative algebra stressing links with algebraic geometry

Can someone recommend some books on commutative algebra stressing links with algebraic geometry? My concern is this. It seems to me that most of commutative algebra was formulated at least initially ...
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### Why is the polynomial $F \in K[Z_0, \dots, Z_n]$ on $K^{n+1}$ not a well defined function on $\mathbb{P}^n$ in general?

I'm reading Joe Harris' Algebraic Geometry and he says "A polynomial $F \in K[Z_0, \dots, Z_n]$ on the vector space $K^{n+1}$ does not define a function on $\mathbb{P}^n$" where $K$ is a ...
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### Image of base change of immersion

I'm revising for my exams now and struggling with the following exercise: Let $f: X \to S$ be an open or closed immersion and $g: S' \to S$ another morphism where $X,S,S'$ are schemes. Then the base-...
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### Genus of Curves over finte fields

This may be a dumb question but is calculating the genus of a curve define over a finite field different than over $\mathbb{C}$. For example the following curve: $y^8 + y +x^{12} + x^5$ is genus 14 ...
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### How to prove that $i^\vee:X^\vee\rightarrow Y^\vee$ is dominant?

In my case $X$ is an abelian variety, $Y$ is an abelian subvariety of $X$, $i: Y\hookrightarrow X$ is the inclusion map,so we have $i^\vee:X^\vee\cong Pic^0X\rightarrow Y^\vee\cong Pic^0Y$, the dual ...
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### The complete solution to a system of polynomials over $\mathbb{R}$

If I am solving a positive-dimensional system of polynomials over $\mathbb{R}$, and specifically am searching only for real solutions, how do I know that my solution is complete and there are no other ...
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### How do you define the restriction of a sheaf?

Just to be clear with the notations: Recall that the pullback of $\mathcal{F}\in\mathcal{O}_B\text{-Mod}$ via $f:A\rightarrow B$ (morphism of schemes) is defined as \begin{equation*}f^*\mathcal{...
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### Bounding the degree of an algebraic extension containing solutions to polynomials

Let $F$ be a field, and let $f_{1},\ldots, f_{s}$ be polynomials in $F[x_{1},\ldots, x_{t}]$. Assume that the degree of the polynomials is bounded by $d$, by which I mean, if $m$ is any term in $f_{i}$...
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### Does an isogeny always define a covering map?

Consider a map $f: G_1 \to G_2$ between two topological groups. If $f$ is an isogeny when viewing $G_1,G_2$ as algebraic groups does $f$ always define a covering map when viewing $G_1,G_2$ as ...
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### Help in showing that the cusp $(y^2-x^3)\subset \mathbb{C}^2$ is not isomorphic to $\mathbb{C}$

Let $X:=(y^2-x^3)\subset \mathbb{C}^2$ be the vanishing of the polynomial $f(x,y)=y^2-x^3.$ I have proved an exercise in Hartshorne: If $\varphi:\mathbb{C} \to X, \ t \mapsto (t^2,t^3)$ is the ...
### Conics and conics of the form $ax^2+by^2+c=0$
The problem of finding rational points on conics is usually discussed (for example in the book of Silverman and Tate) for conics of the form $ax^2+by^2+c=0$. I assume that those conics are in ...