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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Defining invariants of varieties over fields

Let $f$ be a real-valued function on the set of curves over $\overline{\mathbf{Q}}$. Assume that isomorphism curves give rise to the same value for $f$. Let $K$ be a number field and let $X$ be a ...
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122 views

Gaps in the Genera of Space Curves

We learned the following relationship between the degree and genus of plane curves in my algebraic geometry course: \begin{array} a \text{degree} &d &1 &2 &3 &4 &5 &6 ...
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1answer
126 views

Does an inclusion of ring spectra induce a morphism of rings?

My question regards proposition 3.2 in Hartshornes Algebraic Geometry, the statement that a scheme is locally noetherian if and only if for every affine open subset $\operatorname{spec}(A)$, the ring ...
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67 views

Local parameter of a point on a hyperelliptic curve

Let $Y$ be the affine plane curve given by the equation $y^2=F(x)$, where $F$ is a polynomial in one variable of odd degree over a field of characteristic not equal to 2. Let $\xi\in Y$. Suppose ...
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277 views

Computing degree and ramification indices for given morphism of irreducible smooth curves in $\mathbb P^2$

Let $V = Z(X_0^8 + X_1^8 + X_2^8) $ and $W = Z(X_0^4 + X_1^4 + X_2^4)$ in $\mathbb P^2$. It can be shown that $V$ and $W$ are irreducible curves. Let $\phi : V \to W, (X_i) \mapsto (X_i^2)$ be a ...
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68 views

Proving $\mathrm{dim}_k \mathcal{O}_{V,P} / (\pi_P^n) = n$ for $P$ a smooth point with local parameter $\pi_P$ on an irreducible curve $V$

Let $V$ be an irreducible curve with $P \in V$ a smooth point. Let $\pi_P$ be a local parameter of $P$. I'd like to show that $\mathrm{dim}_k \mathcal{O}_{V,P} / (\pi_P^n) = n$ for $n \in \mathbb N$. ...
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192 views

Quotient of an affine variety by a finite group coincides with topological quotient as a point set?

I have just read the construction of the quotient of a closed subset $X$ of affine space by a finite group $G$ of automorphisms of $X$, in Shafarevich, Basic Algebraic Geometry I. Shafarevich gives ...
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129 views

degree of a projective embedding

assume we have an irreducible (even toric in my case) variety $X$ with an ample line bundle $L$. Let d an integer such that $L^d$ is very ample and consider the projective embedding associated to ...
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135 views

Proving that if $X$ is the hypersurface $wx=yz$ in $\mathbb{A}^{4}$ then $X$ is rational.

How do i prove that : $X$ is the hypersurface $wx=yz$ in $\mathbb{A}^{4}$ then $X$ is rational. I do know the definition of $X$ being rational, but don't know how to apply that prove the above ...
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80 views

Understanding a morphism of modules by properties of the induced residue field homomorphism

Let $A$ be a reduced local Noetherian ring, and $\phi: M\to N$ a morphism of finitely generated free $A$-modules. For all $\mathfrak{p}\in\text{Spec}(A)$, let ...
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229 views

Kähler differentials not the same as regular differentials on a singular curve

Let $X$ be the affine cubic curve $y^2=x^3$, over a field of characteristic not equal to 2 or 3. Let $A=k[X]$, the ring of regular functions on $X$. Let $\Omega_A$ be the $A$-module of Kähler ...
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61 views

Finding a non-trivial regular differential form on a singular cubic surface

Let $X$ be the cubic surface $x_0^3=x_1x_2x_3$ in $\mathbb{P}^3$. How to find a non-zero regular differential 2-form on $X$?
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155 views

Closed proper subvarieties of curves are finite sets of points

Why is it true that a closed proper subvariety of a curve is a finite set of points? I had the following lines of thinking: Let $C$ be a curve and $X$ a proper closed subvariety of $C$. Write $C$ as ...
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48 views

Projection of Curve into $\mathbb{P}^{1}$.

Working out some questions from Ravi Vakil's notes. Here is a question: Question: Suppose $\operatorname{char}\bar{k} \neq 2$ and let $C$ be the curve defined by $x^{2}+y^{2} = z^{2}$. Let $\rho$ be ...
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1answer
87 views

Product of projective and affine line is not affine

Why is $\mathbb{P}^{1} \times \mathbb{A}^{1}$ not isomorphic to an affine variety?
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119 views

property of an affine algebraic set in Zariski topology

The following problem is giving me trouble: Suppose $X \subset \mathbb{A}^{n}$ is an affine algebraic set, and $S \subset X$ is a subset. Show that if $\bar{S}$ is the closure of $S$ in the ...
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231 views

How to find the canonical divisor on a nonsingular toric variety?

I am reading Fulton's "Toric Varieties." In it, he explains that if $X$ is a toric variety and if $D_1, \ldots, D_d$ are the irreducible divisors invariant under the big torus action, then $$ ...
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109 views

Number of birational classes of dimension d, geometric genus 0 varieties?

Fix an algebraically closed field $k$ and a positive integer $d$. My question is, what is the number of birational classes of dimension $d$, projective varieties over $k$ with geometric genus 0? If it ...
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475 views

A problem about the twisted cubic

I have some difficulty with the following problem: Let $f : k → k^3$ be the map which associates $(t, t^2, t^3)$ to $t$ and let $C$ be the image of $f$ (the twisted cubic). Show that $C$ is an ...
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125 views

Diagonal in projective space

This is exercise $2.15$ from Harris book "Algebraic Geometry: A First Course". Show that the image of the diagonal in $\mathbb{P}^{n} \times \mathbb{P}^{n}$ under the Segre map is isomorphic to the ...
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How do different definitions of “degree” coincide?

I've recently read about a number of different notions of "degree." Reading over Javier Álvarez' excellent answer for the thousandth time finally prompted me to ask this question: How exactly do ...
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308 views

Problem about Complete Intersection in $\textbf P^n$ (from Hartshorne).

I am in trouble with Exercise 8.4 in Hartshorne's Chapter II; I am doing part (a). It is about (global) complete intersection in $\textbf P^n$. For those without Hartshorne' book at hand, I describe ...
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Krull Dimension of a scheme

Can someone give a hint or a solution for showing that a scheme has Krull dimension $d$ if and only if there is an affine open cover of the scheme such that the Krull dimension of each affine scheme ...
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322 views

Ramification on hyperelliptic curves

I am using Rick Miranda's book "Algebraic curves and Riemann Surfaces" to try and check some things about hyperelliptic curves. I have completed almost all of one of the exercises, but there is one ...
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1answer
143 views

The probability of $Ax^2+Bxy+Cy^2 = 1$ defining an ellipse.

In Keith Kendig's paper, Stalking the Wild Ellipse (published in the American Mathematical Monthly, November 1995), he says that if $A, B, C$ are chosen at random, the probability that the Cartesian ...
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237 views

connections on coherent sheaves

Let $X$ be a smooth variety over $\mathbb{C}$. If $\mathscr{F}$ is a coherent sheaf on $X$ with connection, does it follow that $\mathscr{F}$ is locally free? I can't think of any counterexamples. ...
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271 views

Is this quotient ring $\mathbb{C}[z_{ij}]/\ker\phi$ integrally closed?

A few days ago, I asked a linear algebra question, but it seems that the notions are better stated in terms of algebraic geometry. I don't have much solid knowledge of algebraic geometry, so I'm ...
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2answers
189 views

What if $\operatorname{char}\mathbb{K}$ is not $0$ or if $\mathbb{K}$ is not algebraically closed? (Nullstellensatz)

Given a field $\mathbb{K}$ which is algebraically closed and of characteristic 0, we can say exactly what the maximal ideals of $\mathbb{K}[x_1,\dots,x_n]$ are and they correspond to points in ...
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91 views

Image of $2$-fold map

Show that the image of the veronese map $[a : b] \mapsto [a^{2}: b^{2} : ab]$ is not contained in any hyperplane of $\mathbb{P}^{2}$. Using the result from a previous question I asked: Hypersurface ...
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336 views

What is Hilbert polynomial of this projective variety?

Suppose you have a map $\varphi\colon\mathbb{C}^m\times\mathbb{C}^n\to\mathrm{Mat}_{m,n}(\mathbb{C})$ defined by sending $(\mathbf{u},\mathbf{v})\mapsto\mathbf{u}\cdot\mathbf{v}^T=(u_i,v_j)$. So ...
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1answer
256 views

Hypersurface becomes an hyperplane after embedding

Let $X$ be an hypersurface of degree $k$ in $\mathbb{P}^{n}$, why the equation defining $X$ becomes linear in the Veronese coordinates? More precisely I want to understand the last paragraph of the ...
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1answer
262 views

Resolution of singularities (small resolution)

Consider the following complex (complete intersection) variety, $$ f_1: x_0^2 + x_1^2 + x_2^2 + x_3^2 = 4x_4x_5,$$ $$ f_2: x_4^4 + x_5^4 = 2x_0x_1x_2x_3,$$ in $\mathbb{P}^5$. This is the first example ...
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1answer
184 views

Explicit example of a toric flip

I am looking for a toy example of a flip between toric projective 3-folds. More precisely, I would like to see their defining fans (or polytopes). Does anyone know where I can find something like ...
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432 views

Why morphism between curves is finite?

If $X$ is a complete nonsingular curve over $k$, $Y$ is any curve over $k$, $f: X \to Y$ is a morphism not map to a point (so $f(X)=Y$), then $f$ is a finite morphism. This is the assertion prove in ...
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301 views

Elliptic curves, 2-torsion and branch points.

I'm currently reading through Ravi Vakil's notes on Algebraic Geometry. I've been having trouble grasping some things conceptually though and I hope that you can help me. For an elliptic curve (E,p) ...
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98 views

How to get a geometric morphism out of a section? (And general pedagogy on classifying toposes)

Let $\mathcal{E}$ and $\mathcal{F}$ be toposes, $X$ an object of $\mathcal{E}$ and $p: \mathcal{E}/X \rightarrow \mathcal{E}$ the canonical geometric morphism (whose inverse image part is pullback ...
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915 views

Why is Hodge more difficult than Tate?

There are strong connections between the Hodge and the Tate conjectures, mainly at the level of similarities and analogies. To quote from an answer of Matthew Emerton on MathOverflow: "[...] we ...
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88 views

Closed / embedded surface

Given a closed surface in $\mathbb R^3$, is it necessarily an "embedded surface"? I think it is true, but that is just because I can't think of a closed surface for which we cannot construct a smooth ...
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1answer
192 views

Why is the kernel of this strange polynomial homomorphism what it is?

I've been trying to delve a little further into linear algebra, but I'm not following something I think is supposed to be obvious. Suppose $M_{m,n}(\mathbb{C})$ is the set of rectangular $m\times n$ ...
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1answer
110 views

Varieties given by non-algebraic equations

In algebraic geometry one (mostly) studies varieties given by polynomial equations. Such equations define algebraic varieties and there are many "dictionaries" available. For example, the category ...
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67 views

Automorphism of $L|K$ mapping 3 distinct rational points of $S_{L|K}$ to other 3 distinct ones

Let $K$ be a field and consider $L = K(x)$ the field of rational functions. Let $v_{1}, v_{2}, v_{3}$ rational points in the abstract Riemann surface $S_{L|K}$, distinct from each other, and $w_{1}, ...
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695 views

Finding singular points and computing dimension of tangent spaces (only for the brave)

I'm currently looking at the following two questions: i) Consider $V = Z(I) \subset \mathbb A_k^3$ where $I$ is generated by $X_1^3 - X_3$ and $X_2^2-X_3$. Find the points at which $V$ is singular ...
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224 views

Finding the radical of some ideals

I need to find the radicals of the following ideals: i) $\mathfrak{a} = (xy^3, x(x-y))$ ii) $\mathfrak{b} = (xy^3, x^2(y-3))$ iii) $\mathfrak{c} = (x^2(y-z), xy(y-z), xz(y-z)^2)$ Can I just use ...
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2answers
176 views

Components of a variety in projective space

How do we find the irreducible components of the following projective variety in $\mathbb{P}^{3}$, $V(wy-x^{2},xz-y^{2})$?
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500 views

Function fields of irreducible varieties

The following might seem long, rambling and to contain more information than necessary. The problem is, I'm having a macro-understanding issue and feel like I need to tell you everything I think so ...
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1answer
80 views

Representation of a subset of a finite affine space as a variety

It is simple to see that every subset of a an affine space over a finite field is a variety - for example, it follows from the fact that finite subsets are closed in the Zariski Topology of every ...
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62 views

Continuity of a map of a topological space to a pro-topological space

Let $(X_i)$ be a projective system of topological spaces. Let $X$ be the projective limit of $X_i$. Let $G$ be a topological space. What does it mean for $G\to X$ to be continuous? My guess is that ...
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271 views

Artin-Schreier extensions over characteristic two fields

I have been looking at hyperelliptic curves over an algebraically closed field $k$ of characteristic two, with a view towards finding the basis for the vector space of holomorphic differentials. To do ...
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143 views

Intersection of projective curves

In general, how do we find the intersection of projective curves? For example suppose I have $V_{1}(x^{2}+y^{2}-2z^{2}),V_{2}(x^{2}+y^{2}-z^{2})$ and I want to find $V_{1} \cap V_{2}$ viewed as ...
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83 views

If $Y = X \backslash Z(f)$ for some affine variety $X$ with $p \in Y$, then $T_p Y \cong T_p X$

Let $Y = X \backslash Z(f)$ be a quasi-affine variety for some affine variety $X$, and let $p \in Y$. I'd like to prove that $T_p Y \cong T_p X$. I have the following definition of $T_p X$: If ...