The study of geometric objects defined by polynomial equations. Algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc.

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Algebra, Geometry and Algebraic Geometry

I want to know, what is the difference between Algebra, Geometry and Algebraic Geometry ? Your reply is highly appreciated.
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Image of the Brauer group under a field extension

For $k$ a field, let $Br(k)$ - the Brauer group of $k$ - denote the group of finite-dimensional central simple algebras over $k$, modulo Morita equivalence $(A\equiv B\iff \exists m, n(A\otimes_k ...
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Show that $c-b\lt b-a$

The question: Let $G$ be an Arf semigroup and $a\lt b\lt c$ be three consecutive elements in $G$. How to show that $c-b\lt b-a$ and how to show that this is not necessarily the case for every ...
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Looking for an introductory Algebraic Geometry book

I am looking for recommendations on an AG text to work through this summer, possibly with the help of a mentor. I would want this book to have some introduction to categories, and then develop the ...
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23 views

Computing the genus in positive characteristic

That's an exercise but I'm not so sure how to approach this. Let $k$ be a field of characteristic $p$ and let $f(t)$ be a polynomial in $k[t]$ of degree $d$. Let $C$ be the curve that corresponds to ...
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17 views

Order of $A/2A$ for $A$ an Abelian variety

Let $A$ be an Abelian variety over $\mathbb R$ of dimension $g$. Then the size of $A(\mathbb R)/2A(\mathbb R)$ is $(\# A(\mathbb R)[2])/2^g$. I'm wondering how one might go about proving such a ...
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Zariski closures exercise.

Compute the Zariski clousures $\overline{S} \subset \mathbb{A}^2(\mathbb{Q})$ of the following subsets: (a) $S=\{(n^2,n^3):n \in \mathbb{N}\}\subset \mathbb{A}^2(\mathbb{Q})$; (b) $S=\{(x,y): ...
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Canonical Embedding for Branched Cover

I have the triple branched covering $X$ of $\mathbb{P}^{1}$ defined by $y^{3}=x^{6}-1$. I want to show the following: (i) $X$ has genus 4. (ii) The canonical embedding $\phi: X \rightarrow ...
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26 views

Closed conjugacy classes in $M_n(k)$

Let $k$ be an algebraically closed field, $n$ a positive integer, and consider the action of $\mathrm{GL}_n(k)$ on $M_n(k)$ by conjugation. My professor tells me that semisimple conjugacy classes are ...
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39 views

Criterion to decide the invertibility of polynomial maps

Consider a polynomial map $f:\mathbb{R}^{n-1}\to V\subset\mathbb{R}^n$ where $V$ is $n-1$-dimensional variety in $\mathbb{R}^n$. Are there any conditions on $f$ to determine whether it defines ...
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24 views

Show that there is only one conic passing through the five points $[0:0:1], [0:1:0],[1:0:0],[1:1:1]$ and $[1:2:3]$. Show that it is nonsingular

Show that there is only one conic passing through the five points $[0:0:1], [0:1:0],[1:0:0],[1:1:1]$ and $[1:2:3]$. Show that it is nonsingular
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How to find multiple points and tangent lines at multiple points??

Find multiple points and tangent lines at multiple point of the following curve: $F(X,Y)=Y^3-Y^2+X^3-X^2+3XY^2+3X^2Y+2XY$. Now $F(X,Y)= (X+Y)^3-(X-Y)^2$ $F_x=3(X+Y)^2-2(X-Y)$ $F_y=3(X+Y)^2+2(X-Y)$ ...
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28 views

how to show that $V( Y-X^2 )$ is irreducible?

show that $V( Y-X^2 )$ is irreducible. $Y-X^2$ is an irreducible polynomial ($Y-X^2$ cann't be factored into more irreducible components). Can we conclude that $V(Y-X^2)$ is irreducible??
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Decompose $V(Y^4-X^2,Y^4-X^2Y^2+XY^2-X^3)$ into irreducible components

Decompose $V(Y^4-X^2,Y^4-X^2Y^2+XY^2-X^3)\subset A^2(C)$ into irreducible components. I tried like this: $Y^4-X^2=(Y^2-X)(Y^2+X)$ and $Y^4-X^2Y^2+XY^2-X^3=(Y+X)(Y-X)(Y^2+X)$. What should I do from ...
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An affine plane curve and intersection

Suppose $C$ is an affine plane curve and $L$ is a line in $A^2(k)$, $L \not\subset C$. Suppose $C=V(F), F \in K[X,Y]$ a polynomial of degree $n$. Show that $ L \cap C$ is a finite set of no more than ...
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31 views

$f_*(O_X)=O_Y$ and connectedness of fibers

Suppose $X\to Y $ is a morphism , under what conditions we have direct image sheaf $f_*(O_X)=O_Y$? For example, suppose $\tilde{S}\to S$ is a blow up, do we have $f_*(O_{\tilde{S}})=O_S$? ...
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73 views

Can the dimension of the Zariski tangent space of a complex curve at a singular point be arbitrarily big?

Can the dimension of the Zariski tangent space of a complex curve at a singular point be arbitrarily big ? Is there a formula relating the dimension of the Zariski tangent space and the order of ...
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59 views

Proving the Existence of an Automorphism on $\mathbb{P}^{1}$

I recently came across the following problem while reading: Suppose that a compact Riemann surface $X$ has genus $g>1$. Let $\phi_{i}:X \rightarrow \mathbb{P}^{1}$ for $i=1,2$ be a pair of ...
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The form of subrings of $k[[t]]$

I saw this question in an algebraic geometry book. I tried to solve this. But I did trivial thing, so I don't write what I did here. This is just self-studying. I want to learn how to solve. Please ...
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37 views

pullback of differential form by constant morphism

Let $G/S$ be a group scheme (i'm fine if you want to assume everything is affine). Let $$g : S \to G \in G(S)$$ be an $S$ point of $G$. We can define a morphism $$\phi_g : G \to G$$ defined at the ...
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Holomorphic functions on algebraic curves

I have been asked to solve the following problem, but I really need some help... How are the holomorphic functions $f:C\to D$, where $C,D$ are nonsingular algebraic curves of genus 1? I know that I ...
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Are $R=K[a,b,c,d]/(ad-bc, a^2c-b^3, bd^2-c^3, ac^2-b^2d)$ and $K[x^4,x^3y,xy^3,y^4]$ isomorphic?

I saw somewhere that $R=K[a,b,c,d]/(ad-bc, a^2c-b^3, bd^2-c^3, ac^2-b^2d)$ and $K[x^4,x^3y,xy^3,y^4]$ considered the same. Is it true? Why? I'm a beginner so please answer in details
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20 views

Multiplicity of an affine curve at a point same as that of its projectivization

Consider the projective curve $C=V(P)$ in $\mathbb{P}^2$ where $P(x_0,x_1,x_2)$ is an homogeneous polynomial of degree $d$. At a point $[a,b,1]$, the multiplicity of $C$ is ...
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32 views

Birational Variety

Given a polynomial map $f:\mathbb{R}^2\to V\subset \mathbb{R}^3 $ defined as follows: $$ (z_1,z_2)\mapsto (2z_1-z_2, 2z_1^2-z_2^2, 2z_1^3-z_2^3) $$ This map defines a Variety ($V$) of dimension $2$ in ...
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60 views

Failure of Luroth's theorem for transcendence degree 3

Can somebody give an example which shows the failure of Luroth's theorem for transcendence degree 3 over $\mathbb{C}$
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56 views

A book suggestion - Algebraic geometry. (Arf rings and Hilbert functions)

I am studying algebraic geometry and I need to learn Arf rings and Hilbert functions. Please suggest me books / lecture notes... etc that explains this topic in detail. Thank you.
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44 views

What is the nature of this surface?

What is the nature of the surface whose equation is (it depends on $m$) $$x^2+2y^2+(m+1)z^2+2xy-2yz-2x+2y-4z+m^2+4=0$$
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Bott formula for projective bundles

For a projective space one has Bott formula to compute $h^q(ℙ^n,Ω^p(k))$, where $Ω^p(k)$ is the k-twisted sheaf of sections in the p-th power of the cotangent bundle of $ℙ^n$. I am wondering if there ...
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64 views

Topological invariance of chern classes

Are Chern classes topological invariants? To be more precise: Given two complex manifolds $M$ and $N$. Does a homeomophism $f:M\to N$ map Chern classes to Chern classes?
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66 views

Invertibility of a Polynomial map.

Given following polynomial map $f:\mathbb{R}^2\to V\subset \mathbb{R}^3 $: $$ (z_1,z_2)\mapsto (2z_1-z_2, 2z_1^2-z_2^2, 2z_1^3-z_2^3) $$ Is this map a bijection? If so, how?
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38 views

Two definitions of the Weil restriction.

Let $L/K$ be a galois extension with $G:=\mathrm{Gal}(L/K)$, $X$ a $L$-scheme. We have two definitions of the Weil restriction of $X$ : 1) If the contravariant functor $\mathrm{Res}^L_K(X) : (Sch/K) ...
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Some question on curves

Let $C$ be a nonsingular curve. In the proof of Theorem(V.2.17) of hartshorne book, I see the following statment: We have only to take maps $\mathcal{O} \rightarrow \mathcal{O}(-e)$ and ...
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Tor sheaves on schemes

I was trying to understand the definition of "Tor sheaves", but since it is defined in the derived category of sheaves of $\mathcal{O}_X$-modules and since I am not acquainted with derived categories ...
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32 views

Is the relative ideal of two affine curves $C\subset Z$ a finite dimensional vector space?

Let $I$ be a (non necessarily radical) ideal in the ring $A=\mathbb C[x,y,z]$, with Hilbert function $h=T+n$, where $T$ is a variable and $n>0$ is an integer. Let us assume that $I$ is contained in ...
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55 views

Generalised rigidity lemma

The usual version of the "rigidity lemma" in algebraic geometry says something like this: If $U, \, V, \, W$ are algebraic varieties, with $U$ proper, and $f: U \times V \rightarrow W$ is a morphism ...
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Weak Kodaira Vanishing - Hartshorne III.7.1

In the Serre Duality section of Algebraic Geometry by Robin Hartshorne, the following exercise is posed: If $X$ is an integral projective scheme over a field $k$, prove that an ample invertible sheaf ...
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6 views

How do we construct projective plane using Orthogonal arrays?

Using a OA of strength 2 ( I think) with unit index can we construct a projective plane? In general, how can we construct a incidence geometry using an OA with unit index?
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40 views

Quotient of group schemes and its rational points.

At the moment I have some difficulties in understanding the quotient of group schemes and so exact sequences. I am aware that precise answers would be difficult to be given without speaking of sheaves ...
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Incidence variety fo Grassmmanians

Let $k$ be an algebraic closed field (say, $\text{char}(k)\neq 2$), $n \in \mathbb N\setminus \{0\}$ and $G(m, n) = G(m, \mathbb P^n(k))$ the variety of Grassmmanian of $m$-dimensional linear ...
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Base change by a finite extension

Let $L/K$ be a finite extension of fields Now let $T$ and $X$ be $L$-varieties (we can take $T$ affine if you want) I would like to know if it is true that $$ Hom_L(T\times_K L,X) = Hom_L(T,X)^n $$ ...
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Two continuous functions agree on an open subset of an irreducible space.

I have the varieties $X,Y,Z$ where $X$ is complete. I have the morphism of varieties $f:X\times Y\rightarrow Z$, I have a closed subset $W\subset Y$ such that $f=g\circ\textrm{pr}_Y$ on ...
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22 views

Algebraic closure of a subfield of the field of fraction of a variety

I think that there is the following claim in Basic Algebraic geometry of Shafarevich. Let $X\to Y$ be a dominant morphism of varieties over an alebraically closed field $k$ of $char=0$. Let $\varphi: ...
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61 views

d-uple embedding

When one restricts the $d$-uple embedding $\mathbb{P}^n \hookrightarrow \mathbb{P}^N$ to $\mathbb{P}^{n-1} \hookrightarrow \mathbb{P}^n$, does this yield the $d$-uple embedding $\mathbb{P}^{n-1} ...
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Lefschetz duality for non-compact relative manifolds

I'd like to use the formulation of Lefschetz duality stated here, but I can't seem to find a reference for this particular version of it, and it doesn't seem quite right to me. The exact statement in ...
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Fixed Point Involutions

In recent reading on Riemann surfaces and complex manifolds (primary Miranda with a few random finds online), I encountered the notion of involutions, in particular fixed point involutions. We recall ...
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29 views

Why are there no Dual-octonions?

In the case of quaternions, we can define the traditional quaternions setting the imaginary components equal to root negative one, the hyperbolic quaternions by using root positive one, and the dual ...
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30 views

Irreducible Decomposition of a Vareity

How would one decompose the variety $V=(x+y)^s$ into irreducible components? Would it be just $V=(x+y) \cup (x+y) \cup ... \cup (x+y)$, s number of times? And this is over $\mathbb{C}$.
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Surjectivity and exactness on higher homotopy groups

If $X$ is a normal algebraic variety over $\mathbb{C}$ and $U$ an open set of $X$ then we have surjectivity on the fundamental groups $ \pi_1(U)\rightarrow \pi_1(X)$. If $f\colon X \rightarrow Y$ is ...
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50 views

$L(D)$ is Vector Space

Given a divisor $D$ on a curve $X$, define $L(D)=\{0\}\cup \{f \in k(X),f\ne 0 \, | (f)+D \ge 0\}$. where $(f)=\sum \nu_P(f)P$ and $ \upsilon_{P}(f)= |zeros| − |poles| $ of $f$ at $P$. I want to ...