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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Max Noether's Fundamental Theorem proof in Fulton's book

In the end of the proof we get that $H=A'F+B'G$ and $A'=\sum A'_i$, $B'=\sum B'_i $ while $A'_i$ and $B'_i$ are forms of degree $i$. I don't understand how then he makes the conclusion that $H=A'_sF+B'...
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49 views

Example of a non-Kummer totally tamely ramified Galois extension

Let $A$ be a DVR with fraction field $K$, and let $L$ be a totally tamely ramified finite Galois extension of $K$ of degree $e$ - ie, the integral closure $B$ of $A$ in $L$ is a DVR with ramification ...
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34 views

branched cover of projective line over rationals

I was reading something when I came across the following phrase "branched cover of projective line over rational " . To understand what does author mean , I started reading , Now I know about ...
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1answer
26 views

Kahler differential calculation

Take the conic $X: z^2-xy=0$ in $\mathbb P^2$. On the patch with $z=1$ and coordinates $x, y$, $X$ is cut out by $1-xy=0$. On the patch with $y=1$ and coordinates $u, v$, $X$ is cut out by $v^2-u=0$. ...
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22 views

Tracing the sides of an equilateral triangle

Is there any way I can get the points in 2D plane on the sides of an equilateral triangle for certain infinite animation sequence? For example in case of tracing the circumference of the circle, I ...
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48 views

When is a finite $R$-algebra isomorphic to $R$?

Let $R$ be a $\bar{k}$-algebra (of finite type or complete) reduced (and maybe integral, if needed), let $A$ be an $R$-algebra, finite as an $R$-module, reduced and connected and such that there ...
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1answer
22 views

Sections of a finite étale cover of a connected scheme which coincide at a geometric point

Let $\phi_1, \phi_2 : S \longrightarrow X$ be two sections of a finite étale cover $X \longrightarrow S$ of a connected scheme $S$. Assume that $\phi_1 \circ \overline{s} = \phi_2 \circ \overline{s}$ ...
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118 views

Why is $\mathbb{C}^2\setminus\{(0,0)\}$ not a basic open set?

Consider the affine variety $\mathbb{C}^2$ equipped with Zariski topology. By the question above, I mean why $X:=\mathbb{C}^2\setminus\{(0,0)\}$ cannot be written as $$ X:=U_f:= \{(x,y)\in \mathbb{C}...
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42 views

A cartesian diagram?

let $k$ be a field, and $X$ and $Y$ varieties over $k$. Let $L$ be an extension of $k$, and $X_L=X\times_k L$. Is the diagram $$\require{AMScd} \begin{CD} X_L\times Y_L\times X_L @>>> X\...
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44 views

Making sense out of $F$-structures and the notion of $F$-variety

For almost two years I have been trying to make sense out of several claims about varieties over nonalgebraically closed fields made in the first chapter of the textbook Linear Algebraic Groups by T.A....
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52 views

Quick question: Extension of vector bundles on a compact Riemann surface

Given the following short exact sequence of holomorphic vector bundles on a compact Riemann surface: $0\rightarrow M\rightarrow E \rightarrow N\rightarrow 0$ Fix a hermitian metric on $E$ and $n=...
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119 views

What exactly is $\mathbb{P}_\mathbb{Z}^n$?

So, I have the following definition of $\mathbb{P}_A^n$ for an arbitrary (commutative) ring $A$, from Hartshorne: Set $S=A[x_0,\ldots,x_n]$, so that $S=\bigoplus_{d\geq 0}S_d$ as a graded ring, $S_+=\...
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1answer
55 views

About the definition of polynomials on vector spaces?

In the book Linear Systems Theory and Introductory Algebraic Geometry (R. Hermann) the author defines A polinomial on V (a $\mathbb K$-vector space) is an element of the smallest ...
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63 views

example of toric varieties with nontrivial first cohomology group

If I remember correctly, when a toric variety is smooth or simplical (the moment polytope is simplicial rational), then there is no odd dimensional cohomology group and the dimension of the even ...
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26 views

A question of terminology regarding exceptional curve or is it divisor.

So I kept on reading the book by Griffiths and Harris called Principles of Algebraic Geometry and I've seen a definition of exceptional divisor of the first kind. On page 487: A smooth rational ...
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91 views

Is there an elliptic curve with exactly one rational point?

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. Is there an example of such an $E$ such that the only rational point in $E(\mathbb{Q})$ is the point at infinity? In other words, consider the ...
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28 views

must this extension of a DVR be unramified?

Let $A$ be a normal domain, and $P$ a height 1 prime, then $A_P$ is a DVR. Let $K$ be the fraction field of $A_P$, and let $L$ be a finite Galois extension of $K$ of degree $e$, let $B$ be the ...
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55 views

Showing $\exp:\mathscr{O}_X\to\mathscr{O}_X^*$ is an epimorphism of sheaves

$\newcommand{\O}{\mathscr{O}}$Let $X=\Bbb{C}$. Define $\O_X$ to be the sheaf of holomorphic functions, and $\O_X^*$ to be the sheaf of invertible (i.e. nonvanishing) holomorphic functions, the latter ...
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1answer
54 views

Why does this homological lemma hold?

Let $A$ be a noetherian ring; let $C^{\boldsymbol\cdot}$ be a bounded above complex of flat $A$-modules in positive degrees, let $L^{\boldsymbol\cdot}$ be a bounded above complex of free $A$-modules ...
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37 views

Are local rings of non-singular curves noetherian integral domains?

Let $P$ be a point on a nonsingular curve $Y$, then the local ring $\mathcal{O}_P$ is a regular local ring of dimension one. Hartshorne gives the following theorem: Let $A$ be a noetherian ...
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68 views

Loci of intersection points of two curves

There are two continuous, negatively-sloped curves,A and B. They intersect at least once ,say at $(x,y)$. If I introduce a third curve C, whose X axis intercept has a higher magnitude than that of B, ...
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29 views

Quotient field of ring of regular functions at some point in affine variety is the field of rational functions on the variety

I am reading Hartshorne's book of Algebraic Geometry. I am stuck in understanding why quotient field of the local domain O_p (where O_p denotes the ring of regular functions at a point p in affine ...
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33 views

Multiplicity of Cartier divisor on locally noetherian scheme is only non-zero at generic point

I'm following chapter 7 in Qing Liu's book 'Algebraic Geometry and Arithmetic Curves' about 'Divisors and applications to curves'. My question concerns Definition 1.27: Let $A$ be a Noetherian ...
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75 views

A question on connectedness

I was going through a basic problem book in algebraic geometry. There in the very first chapter I have encountered a problem which asks to prove that hyperbolas are connected in $\mathbb{C}\times \...
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18 views

Connection between saturated ideals an CM algebras.

Let $I$ be an homogenous ideal of the polynomial ring $K[x_1,\dots,x_n]$. Is there any relations between $I$ being saturated and $R/I$ being a Cohen-Macaulay?
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133 views

Why is $\mathrm{Spec}(\mathbb{Z})$ a terminal object in the category of affine schemes?

I've seen this claim repeated in many places (always without source or proof), that $\mathrm{Spec}(\mathbb{Z})$ is a terminal object – however, the most I've been able to prove myself is that for any ...
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1answer
60 views

Intermediate cohomologies vanish in complete intersections (Hartshorne ex. III.5.5)

I have problems proving the following assertion: Let $X=\mathbb{P}^n$ be the projective $n$-scheme over some field $k$, and $Y \hookrightarrow X$ a complete intersection, i.e., pure $q$-dimensional ...
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50 views

Geometric interpretation of a result from commutative algebra

I have come across the following result in Hartshorne, $I.6.5$ for those who have the book. The result says that if $K$ is a finitely generated extension of some base (algebraically closed) field $k$ ...
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56 views

Combinatorial question relating to zero sets of ideals

Let $R$ be a ring and $I$ an ideal of $R[x_1,\ldots,x_n]$. Then define the Zariski closed set $$V=\{x\in R^n:f(x)=0\text{ for all }f\in I\}.$$ I'm interested in the quantity $$p(f)=\frac{|\{x\in V:f(x)...
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24 views

Is the normalization of the cusp $H$-projective?

I have a slight confusion about a statement I think to be true and if the normalization of the cusp is "good enough", this would sadly provide a counterexample. So here is my hope, that the ...
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53 views

Canonical sheaf of product

This is a follows up to Canonical divisor of product of varieties. I am in the case where $X$ is a smooth variety over a field $k$, and $K$ an algebraic closure of $k$. How can i express the ...
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51 views

Fully faithful functors between schemes

Let $f:X\rightarrow Y$ be a proper morphism of schemes (satsfying some technical conditions), and $f_\ast$ the functor from the category of coherent sheaves on $X$ to coherent sheaves on $Y$ (by ...
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40 views

Why is axiom of dependent choice necesary here? (noetherian space implies quasicompact)

In the proof of noetherian space implies quasicompact I am reading, it goes as follows: Let $X$ be a noetherian space. Let $U$ be the collection of open subsets of $X$ that can be expressed as a ...
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45 views

Why only generic?

I have the following situation: Let $X$ and $Y$ be two projective irreducible varieties with the same dimension and $f:X\to Y$ a surjective morphism between them. Then $f$ is generically finite. Why ...
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52 views

Fundamental group of the unit tangent bundle on the genus 2 torus?

I'm interested in the 3-dimensional model geometries; specifically $\widetilde{SL}(2,\mathbb{R})$. I'm looking for a good (see, easily visualizable) example of a compact manifold formed as a quotient ...
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1answer
55 views

Comparing the prime spectra of $\mathbb{Q}[x],\mathbb{R}[x]$ and $\mathbb{C}[x]$

I understand that $\mathrm{Spec}\mathbb{Q}[x]=\{(0),(f(x)): f(x)\mbox{ is an irreducible polynomial}\}.$ An argument is the following one: $(0)$ is prime because $\mathbb{Q}[x]$ is an integral domain. ...
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1answer
62 views

Blow-up of a point on a smooth simply connected variety

Apologies if this is an obvious question. Suppose I have a variety which I know is smooth and simply connected and blow-up a smooth point so that the resulting variety is smooth. Does the exact point ...
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92 views
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Projection of a curve in $\mathbb{P}^3$

I've been reading the proof of Theorem IV.3.10 in Hartshorne (p. 313 - 314), which states the following: Given a curve $X \subset \mathbb{P}^3$, there is a point $O \notin X$ such that the ...
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78 views

Quotients of Elliptic Curves

I am fairly inexperienced with elliptic curves so there might be aspects of my question that may need better wording but let me know if there are any issues: Question: Say I have an elliptic curve ...
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1answer
35 views

Finding cohomology group of open dense subset of Schubert variety

Let $Y=Gr_{m}(\mathbb{C}^n)$ be the Grassmannian of $m$-plane inside $\mathbb{C}^n$. Let $X$ and $X'$ be two Schubert varieties inside $Y$ such that $X'\subset X$ and $dim(X')<dim(X)$. Let $Z=X\...
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60 views

When does the Grothendieck spectral sequence converge?

I am trying to understand spectral sequences in algebraic geometry. One has the Grothendieck spectral sequence for composition of functors $\mathcal F: \mathcal A \to \mathcal B$, and $\mathcal G: \...
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56 views

Elementary question: local computation of curvature on principal bundle

Let $G$ be a Lie group and $S=[0,1]^2$. Let $\omega$ be a connection $1$-form on the trivial principal $G-$bundle $P=S\times G$ over $S$. Let $(x_1,x_2)$ be coordinates on the base $S$. We can choose ...
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63 views

Determining if Algebraic varieties are homeomorphic

So attempting to use the language of algebraic geometry, and algebraic variety $V$ is a the set of points that is the solution to some collection of algebraic relations $$x_1, \ldots, x_n \ \text{s.t....
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19 views

What are minimal paths, generators of graph ideal in a cyclic graph $C_n$?

Minimal cuts are the generators of the cut ideal while the Alexander duality of path ideal generated by the minimal paths is the cut ideal -- more on Graph ideals here. Graph ideals are special case ...
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27 views

Add $P$ to itself $N$ times on elliptic curve $y^2 = f(x)$, end up with expression in denominator of $x$ vanishing iff $NP$ is point at infinity?

See the second to last paragraph from page 39 of Koblitz's Introduction to Elliptic Curves and Modular Forms. Why is it that when we add a point $P$ to itself $N$ times on an elliptic curve $y^2 = ...
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36 views

Singular Locus of a Schubert variety

I am trying to compute the singular locus of the schubert variety $X_w$ in $G_{2,7}$ where $w=(4,7) \in I_{2,7}$. Following the notation in the book "The Grassmannian Variety: Geometric and ...
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1answer
59 views

Which affine schemes are projective?

Let $k$ be a field. Are there any useful necessary and sufficient conditions on $k$-algebras $A$ such that $\mathrm{Spec}(A)$ is a projective scheme over $k$? I know that there are very few of these, ...
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51 views

Understanding the prime ideals in the ring of dual numbers over a field

I want to understand $\mathrm{Spec}(k[\epsilon]/(\epsilon^2))$ where $k$ is an algebraically closed field and $R:=k[\epsilon]/(\epsilon^2)$ is the ring of dual numbers. Here is my attempt: Every ...
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1answer
56 views

Irreducibility of a quadric

I am struggling with a problem in Shafarevich's Basic Algebraic Geometry. First, some context: Fix $k$ an algebraically closed field. Lines in $\mathbb{P}^3$ correspond to planes through the origin in ...
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41 views

Calculating the sheaf of relative differentials

I'm trying to understand the sheaf of relative differentials $\Omega_{X/Y}$ with a given morphism of schemes $f:X\to Y$. To make things concrete, I want to consider the example of a parabola ...