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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A further question on the irrationality of $x^2+y^2=3$

(Apologies for a further question on the same problem) On page 79 of Julian Harvil's book "The Irrationals" he sets out to prove (by contradiction) that all the points on the circle described by ...
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72 views

Character Sums, Weights, and Cohomology

I know, vaguely, that certain bounds for character sums over finite fields can be determined by looking at weights and the dimensions of certain compactly supported cohomology groups. However, I've ...
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44 views

Fibres in a power series ring versus fibres in a polynomial ring (a simple question)

Let $A$ be a commutative ring and $p \in \operatorname{Spec} A$. In Matsumura's Commutative Ring Theory p. 118 it is mentioned that even though $A[x] \otimes \kappa(p) = \kappa(p)[x]$, it is not ...
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156 views

What is the stalk at a point of the quotient of a scheme?

This came up when I was talking to @Benjalim on the chat. Consider the space $X=\operatorname{Spec} \mathbb C[x]$. With the usual structure sheaf, this is a scheme. Let $Y$ be $X$ with the points ...
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188 views

Why are constructible sets a disjoint unions of locally closed sets

Let $X$ be a noetherian scheme. The constructible sets are the smallest boolean algebra containing all of the open sets. It is easy to see that the constructible sets are exactly finite unions of ...
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518 views

The projection formula in Algebraic Geometry

In this book, the projection formula stated as follows; Let $f:X\to Y$ a separated, quasi-compact morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on X, $\mathcal{G}$ be a ...
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104 views

Serre's criterion for affinity

I am studying Algebraic Geometry with this book. Through the Serre's criterion for affinity, we can know that $H^1(X,\mathcal{F})=0$ then $X$ is affine with some conditions on $X$ and $\mathcal{F}$. ...
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177 views

How many types of surface singularities multiplicity two exist?

All varieties are over $\mathbb{C}$. Let $S$ be a reduced algebraic surface in $\mathbb{P}^3$ with a singular point $p$ of multiplicity two. The question is local so we reduce to $S \subset ...
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117 views

A zero for a homogeneous polynomial is a zero for the associated inhomogeneous polynomial

I am trying to prove a simple statement from Reid, Undergraduate Algebraic Geometry, pg 16. Let $F(U,V)$ be a nonzero homogeneous polynomial of degree $d$: ...
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220 views

The analytic and the algebraic “small disc”

I would like to understand the relation between an analytic object (the so called "small disc") and an algebraic one (the spectrum of a DVR). The framework is that of one-parameter families of complex ...
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357 views

What are the equations for the image of an algebraically defined subset under the Segre embedding?

Let $\psi: \mathbb{P}^r \times \mathbb{P}^s \to \mathbb{P}^N$ be the Segre embedding with $N = rs + r + s$, as in Hartshorne exercise I.2.14. To be explicit: the image of the pair $([a_0 : \ldots : ...
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135 views

algebraic geometry exercise: infinite subset is dense

A hypersurface $C \subset \mathbb{A}^2$ we call a plane curve. Show that any infinite subset of an irreducible plane curve $C \subset \mathbb{A}^2$ is dense in $C.$ Note. We call hypersurface the ...
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165 views

A question on an answer on Math Overflow about the tangent bundle

I have a question on the accepted answer of this Math Overflow question. Let $K$ be a field and $X$ a $K$-scheme. Define the morphism of schemes $T=\operatorname{Spec}\operatorname{Sym}(\Omega_{X/K}) ...
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124 views

Why is this group action a morphism of varieties?

I am examining a proof that any affine connected algebraic group is a closed subgroup of some $GL_n$, and I am stuck on a fine point. Let $G$ be an affine algebraic group in the sense that it is a ...
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136 views

Triviality of vector bundles

Let $X$ be a proper curve, not necessarily smooth nor reduced, and $E$ a vector bundle on $X$ of rank $r$. Assume we know that $H^0(X,E)\geq r$ and $H^0(X,E^{\vee})\geq r$, can we conclude that $E$ is ...
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175 views

Exercise 4.30 of Eisenbud

I am doing exercise 4.30 from Eisenbud's Commutative Algebra With A View Towards Algebraic Geometry which I append here: Exercise 4.30: Suppose $k$ is a Noetherian ring and for every finitely ...
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117 views

Characters of diagonalizable algebraic groups with no p-torsion

Let $G$ be a diagonalizable algebraic group and $X$ be the character group of $G$. Let $Y$ be a subgroup of $X$. We define $Y^{\perp}$ to be all the $x\in G$ such that $\chi(x)=1$ for all $\chi\in Y$. ...
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139 views

Commutative ring and its group-algebra, and abelian-group-algebra as a commutative ring.

In course of discussing the algebraic structures, one of my seniors is led quite naturally to considering the $\color{red} {geometric}$ version of following: Question: Since we could consider the ...
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193 views

Question on Segre embedding

This is in fact a problem in GTM52 of Hartshone. Define $\psi : \mathbb{P}^{n}\times \mathbb{P}^{m}\longrightarrow \mathbb{P}^{N}$ where $N=rs+r+s$ by $(a_0,...,a_r)\times ...
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300 views

Proving that the differential on an elliptic curve $E$ given by $\omega=\frac{dx}{y}$ is translation invariant

I'm taking a course on elliptic curves and I'm stuck on a line in a proof. We're assuming we're in an algebraically closed field $K$ and char($K)\not=2$. We have our elliptic curve ...
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158 views

Easy to state high-dimensional consequences of Bezout theorem

A classical consequence of Bezout's theorem for plane curves is Pascal's theorem. I am curious if there are some other statements that you find pretty that can be formulated (almost) as elementarily ...
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130 views

Solutions of $\frac{1}{\cos \theta} = a \sin \theta - b$

One of my math professors and I are working on a physics problem involving spinning a chain, and we decided to go as simple as possible and work out the solution explicitly for that case (a long rod ...
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237 views

Does the singular locus of a conical variety (or scheme) determine the singular locus of its projectivization?

Lets say $X$ is a conical affine algebraic variety (conical meaning $X$ is the zero set of homogeneous polynomials of positive degree, equivalenty $X \subsetneq k^n$ and $x \in X \Rightarrow ax \in X$ ...
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89 views

What is Weil paring computing really?

I have trouble in understanding Weil paring on $N$-torsion points on an elliptic curve. Please see Wikipedia for the definition of Weil paring. I would like to know what Weil paring is computing ...
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694 views

Equivariant sheaves

What is the best introduction (textbook) to equivariant sheaves on algebraic varieties equipped with an action of an algebraic group?
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130 views

divisors and powers of line bundles

Can anyone help me with the following question? Let $X$ be a smooth, projective algebraic variety. Let $D$ be an effective divisor on $X$ and $m$ an integer. Under which conditions there exists a ...
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168 views

About prime ideals partial derivatives of polynomials

Given a polynomial $f(x_1,... ,x_n)\in \mathbb{C}[x_1, ... ,x_n]$, we can formulate its (formal) partial derivative with respect to each of the $x_i$, say $f_{i}$. If $f\in \mathfrak{p}$ and $f_{i}\in ...
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133 views

What is the tangent space for intersection point of irreducible varieties.

Given a (real) plane, I know the definition of tangent space for irreducible varieities. (If $P \in V(f_1, \cdots, f_n)$ irreducible, then $T_P V = V(f^{(1)}_{1,P},\cdots, f^{(1)}_{n,P})$. Suppose ...
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108 views

Universal parametrization for orthogonal matrices

Let $k$ be a field whose characteristic is zero and let $n\geq 1$. Say that a matrix $M\in {\cal M}_{n\times n}(k)$ is almost orthogonal if $M^{T}M$ is a nonzero multiple of the identity. Denote the ...
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359 views

Algebraic independence and dimension of a variety

A set of polynomials $\{f_1,\ldots,f_m\}$ in $k[x_1,\ldots,x_n]$ are algebraically independent over $k$ iff for all polynomials $p \in k[y_1,\ldots,y_m]$, $p(f_1,\ldots,f_m) = 0$ implies that $p = 0$. ...
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167 views

Algebraic definition of blow-ups

Let $X$ be a scheme. Choose $C\subset X$ be a subscheme of $X$ and let $\mathcal{I}\subset \mathcal{O}$ be the corresponding ideal sheaf. Then $\mathcal{B}=\oplus_{d\ge0}\mathcal{I}^d$ is a sheaf of ...
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250 views

About first Chern class and Poincaré duality in case of an ample divisor

Let $D$ be a very ample divisor in $X$ projective variety. I can't understand why the first chern class $c_1(\mathscr{O}_X(D))$ equals the Poincarè dual of D, $\mathscr{P}(D)$
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173 views

Dr. Math on factoring - mistake?

I am reading this article http://mathforum.org/library/drmath/view/75056.html and would like to ask if this section is correct: If it can, then you would have $f(x,y) = g(x,y) * h(x,y)$, ...
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203 views

Primes of good reduction for varieties

Suppose I have a (smooth, projective) variety $X/\mathbb{Q}$. Is the notion of a primes of bad/good reduction well-defined from this data? Motivation and attempt at an answer: The question should be ...
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144 views

Ring homomorphism and affine scheme

How to describe all ring homomorphisms $f: A \rightarrow B$, such that corresponding affine scheme morphism $f: Spec \, B \rightarrow Spec \, A$ is open immersion?
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543 views

Motivation for studying quadratic algebras, Koszul algebras, Koszul duality

I'm trying to gain a practical understanding of Koszul duality in different areas of mathematics. Searching the internet, there's lots of homological characterisations and explanations one finds, but ...
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156 views

If the special fiber of a flat morphism is reduced, then any other fiber is reduced?

Suppose $R=\mathbb{C}[x_1,\ldots, x_n]$ is a polynomial ring with $I$ being an ideal of $R$. Let $I'$ be an ideal of $R[t]$. If $R[t]/I'$ is flat as a $\mathbb{C}[t]$-module and over $0$, ...
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110 views

Definition of S(n) for graded ring S

In Hartshorne's Algebraic Geometry, the twisting sheaf of Serre $\mathscr{O}(n)$ is defined to be $S(n)^\sim$, where $S$ is an $\mathbb{N}$-graded ring. But I couldn't find the definition of $S(n)$ ...
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504 views

Literature on the hyperelliptic involution

I'm currently trying to familiarize myself with reducible Jacobians of hyperelliptic curves. A construction which I recently saw in a paper was the quotient $C/\tau$ of a curve $C$ of genus $2$ by a ...
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71 views

some question of combination

we know that hilbert seris of n- variables polynomial ring is $\Sigma_{i} \binom{n-1+i}{i}t^{i}$ But, I don't know $\Sigma_{i} \binom{n-1+i}{i}t^{i}=(1-t)^{-n}$. I wonder to prove in detail.
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280 views

Blow-up along an ideal sheaf

Let $k^2=\operatorname{Spec} \; k[x,y]$ where $k$ is an algebraically closed field. Let $\mathcal{I}$ be the ideal sheaf defined by $(x,y)$. Then $$ Bl_{\mathcal{I}}k^2 $$ is covered by two open ...
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497 views

references for singularities: does quotient singularities imply gorenstein?

Is there a good place where to learn about singularities of algebraic varieties? OK, this question is terribly vague, so I'll ask what I'm currently interested in: if X is a smooth variety and G is a ...
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148 views

Ramification of a prime in a Dedekind ring and curves

Let $\phi:C_1\to C_2$ be a nonconstant map of two smooth curves over some algebraically closed field $K$ and let $P\in C_1$. $\phi$ gives us an induced map of fields $\phi^*:K(C_2)\to K(C_1)$, ...
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166 views

Smoothness over a field and regularity

Hartshorne, Algebraic Geometry In example III.10.0.3, Hartshorne remarks that with k algebraically closed, X smooth of dimension n over Spec k is equivalent to X regular of dimension n. He ...
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229 views

discriminant of an étale cover of an elliptic curve

Let $\pi:X\to E$ be a finite étale morphism, where $E$ is an elliptic curve over a number field $K$. Assume $X$ to be connected, and to be of genus 1. Edit: Assume $X$ and $E$ have semi-stable ...
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157 views

D-Modules and Connections

Let $X$ be a smooth algebraic variety with tangent sheaf $\Theta$ and $M$ a sheaf of modules on $X$. Let $D$ be the sheaf of differential operators on $X$. Then giving a left $D-$ module structure on ...
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226 views

For any irreducible variety $X$ and any point $x$ in $X$, dim$\mathscr{T}(X)_x \geq$ dim$X$, with equality holds in a dense open subset of $X$

For any irreducible variety $X$ and any point $x$ in $X$, $\mathrm{dim}\mathscr{T}(X)_x \geq \mathrm{dim}X$, with equality holds in a dense open subset of $X$. Here, $\mathscr{T}(X)_x$ denotes ...
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189 views

Subrings of formal series rings

Let $k$ be a field and $A = k[[x_1, \dots, x_n ]]$ be the ring of formal series in $n$ variables. Consider $g_1, \dots, g_m \in A$ such that $g_1(0) = \cdots = g_m(0) = 0$. For every $f \in k[[t_1, ...
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349 views

A question on an exercise in Fulton's book Algebraic curves

Is there a neighborhood of $(0,0,0)$ on $V(x^2-y^3, y^2-z^3)$ that is isomorphic to an open subvariety of a plane curve?
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What are the sections of $\mathbb{C}\rightarrow \mathbb{C}[[x,y]]^{alg}[\frac{1}{xy}]$?

I've been thinking about weird rings recently, and I couldn't answer the following question to myself: What are the sections of the inclusion $\mathbb{C}\rightarrow ...