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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108 views

Is there any visual animation to show the basic concept of algebraic geometry? [closed]

Is there any visual animation to show the basic concept of algebraic geometry? There are rarely pictures in textbooks, so are there any animation to show basic but important concepts?
4
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1answer
89 views

What is $\operatorname{Pic}(\mathbb{P}^n_{\mathbb{Z}})$?

I would like to know the Picard group of the projective spaces over the integers $\mathbb{Z}$. I know that the projective space over a field $k$ has $\operatorname{Pic}(\mathbb{P}^n_{\mathbb{k}}) \...
3
votes
1answer
195 views

rank of quadrics

Consider the quadric $xw-yz$ in $\mathbf{P}^3$ (all over $\mathbf{C}$), and the Klein quadric $x_0 x_5+x_1 x_4+x_2 x_3$ in $\mathbf{P}^5$. I want to determine the rank of these quadrics. For the first ...
9
votes
1answer
493 views

Is there a coherent sheaf which is not a quotient of locally free sheaf?

Suppose $X$ is an algebraic variety, is there a coherent sheaf $\mathcal{F}$ on $X$ which is not a quotient of locally free sheaf? (Hartshorne II Cor 5.18 showed that on every projective variety, ...
3
votes
2answers
102 views

on the coordinate ring of $\mathbb{A}^n \times \mathbb{P}^{m}$

Consider the product $\mathbb{A}^n \times \mathbb{P}^{m}$. Let $x_i$ be affine coordinates on $\mathbb{A}^n$ and $y_j$ homogeneous coordinates on $\mathbb{P}^{m}$. Question: Is $A=k[x_1,\dots,...
2
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1answer
166 views

a topological question regarding the blowing-up of a line

The context of this question is argument (3) in the blow-up section p.28 in Hartshorne. All necessary details are given. Let $x_1,\dots,x_n$ be affine coordinates for $\mathbb{A}^n$ and $y_1,\dots,...
0
votes
0answers
118 views

Book Suggestion - Complex algebraic surfaces

I am studying for an exam of algebraic geometry, in particular, I am dealing with ruled surfaces and numerical invariants, rational surfaces, Castelnuovo's Theorem and its application. I am reading ...
0
votes
2answers
56 views

Parabolas and axis of symmetry?

I have the parabola $$(x+y)^2 = 8(x−y)$$ and know that the axis of symmetry is $$x+y=0$$ but I know when this is the case the left hand side equals 0 but apart from that I can't see how this equation ...
1
vote
2answers
115 views

Is equation for ellipse in polar coordinates correct?

Wikipedia gives the following equation for the conic sections in the polar coordinate system: $r = \frac{l}{1+e\cos\varphi}$. According to the article on conic sections, in case of an ellipse $e = \...
0
votes
1answer
100 views

Generalisation of a result on Kahler differentials

Let $B$ be a local ring which contains a field $k$ of characteristic zero, isomorphic to its residue field $B/\mathfrak{m}$. We know that the map $\delta:\mathfrak{m}/\mathfrak{m}^2 \to \Omega^1_{B/k} ...
2
votes
1answer
89 views

Disconnected Algebraic Set over non-Algebraically Closed Field

I'm trying to find an algebraic set $V$ that can be written as the disjoint union of two proper algebraic sets, such that the coordinate ring $k[V]$, where $k$ is NOT algebraically closed, is NOT the ...
0
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0answers
44 views

Criterion for disconnectedness of affine algebraic set.

I'm trying to prove that, if $V$ is an affine algebraic set, then $V$ is connected in the Zariski topology iff $k[V]$ is not the direct sum of two ideals. Note that $k$ is algebraically closed here. ...
1
vote
2answers
38 views

Do the zeros of a prime ideal on closed points of the Zariski topology uniquely determine it?

That is, if two prime ideals share the exact same zeros on maximal ideals, are they the same ideal? Or at least is there a result with other assumptions that shows this? Learning algebraic ...
2
votes
2answers
108 views

An extension of line bundles splits locally

Consider an extension $0\rightarrow L \overset{\alpha}{\rightarrow} E \overset{\beta}{\rightarrow} L' \rightarrow 0$ of bundles and bundle homomorphisms, where $L$ and $L'$ are line bundles. (Let's ...
2
votes
2answers
77 views

Smoothness and field of fractions

If $k$ is an integral domain and $A$ is a Noetherian finitely presented $k$-algebra for which $A \otimes_k Q(k)$ is a smooth $Q(k)$ algebra, then can it be deduced that $A$ was initially smooth over $...
2
votes
0answers
134 views

Local complete intersection scheme, conormal sheaves and differentials

Let $X$ be a smooth projective variety over $\mathbb{C}$ and $Z \subset X$ be a local complete intersection subscheme in $X$. Denote by $I_Z$ the ideal sheaf of $Z$ in $X$ and $\Omega^1_X$ the sheaf ...
3
votes
1answer
109 views

Isomorphism between Ext groups in Huybrechts and Lehn's book Geometry of Moduli Spaces of Sheaves

On p.46 (or p. 43 in the 1st edition) of Huybrechts and Lehn book Geometry of Moduli Spaces of Sheaves, 2nd ed., they write: Since $K$ is $A$-flat and $I \otimes_k F_0$ is annilated by $m_A$, ...
5
votes
1answer
88 views

Hilbert Nullstellensatz and ring of continuous functions

Is there any relation between Hilbert's Nullstellensatz and the fact that the maximal ideals in $\mathcal C([0,1])$ correspond to a point in $[0,1]$ (which can be generalized to compact hausdorff ...
1
vote
1answer
51 views

What is $\overline{Y}$ in $\text{Spec}A$?

Consider a subset $Y$ of $\text{Spec}(A)$. (Here $A$ is a commutative ring.) What is the closure of $Y$ (or $\overline{Y}$)? I have been under the impression that $\overline{Y}$ is the set of prime ...
1
vote
2answers
196 views

1-form on Riemann Surface

Good evening, I can not prove the following result: Let $\omega $ be a meromorphic 1-form on $ \mathbb {C} _ {\infty} = \mathbb {C} \cup \infty $ such that $ \omega_{|\mathbb{C}} = f (z) dz $. Show ...
2
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0answers
63 views

Regular elements of a module is open and dense

Let $\mathfrak{g}$ be an algebraic Lie algebra and $V$ be a $\mathfrak{g}$-module, then for each $v\in V$, define $\mathfrak{g}^v = \{x\in\mathfrak{g}:xv = 0\}$. Let $V_{reg}$ be the set of all $v$ ...
17
votes
2answers
966 views

History of Algebraic Geometry: Motivation behind definition of schemes

I am trying to read an article by Jean Dieudonne which talks about development of Algebraic Geometry. The article was being published in the journal "Advances in Mathematics" Volume 3, Issue 3, Pages ...
12
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1answer
320 views

Hartshorne Theorem 8.17

I can't understand the proof of theorem 8.17 from Hartshorne's "Algebraic Geometry". Namely, he says that we have an exact sequence $$ 0 \to \mathcal J'/\mathcal J'^2 \to \Omega_{X/k} \otimes \...
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0answers
99 views

Hartshorne Problem I.3.20

Problem I.3.20 in Hartshorne asks to show that if $Y$ is a variety such that $\dim Y \ge 2$ and $Y$ is normal at a point $P$, then any regular function on $Y-P$ extends to a regular function on $Y$. I ...
0
votes
1answer
47 views

an apparent contradiction regarding the local ring at a point

I have encountered an apparent contradiction: Let $Y$ be an affine variety of $\mathbb{A}^n$ and $P$ a point of $Y$. Then i have proved that $\mathcal{O}_P$ is an integral domain and it is also not an ...
3
votes
1answer
62 views

Surjective étale morphisms on points. [closed]

Let $X$ and $Y$ be schemes over a field $K$. We assume, moreover, $X$ and $Y$ to be of finite type, separated and geometrically integral. Let $f:X \rightarrow Y$ be a surjective étale morphism. Is it ...
0
votes
1answer
41 views

Question about toric ideal

In the proposition 1.2 contained in the following http://www.math.harvard.edu/~jbland/ma232b2_notes.pdf, I can't understand why a monomial satisfying (1.7) exists. Can you help me? Thanks.
0
votes
0answers
51 views

What is $T^*(\mathbb{A}^1)$?

Let $\mathbb{A}^1$ be the affine line and $T^*(\mathbb{A}^1)$ the contangent space of $\mathbb{A}^1$. What is $T^*(\mathbb{A}^1)$? Is $T^*(\mathbb{A}^1) = \mathbb{A}^2$? Thank you very much.
1
vote
1answer
133 views

Constructible sets

Is it possible to write down all the constructible sets in $\mathbf{C}$ (endowed with the Zariski topology) or some other "simple" space?
2
votes
2answers
103 views

Is algebraic closure necessary? (3.6.K Ravi Vakil's notes)

I've just done exercise 3.6.K in Ravi Vakil's notes and noticed that my solution does not seem to rely on algebraic closure, so I'd like a sanity check. I understand it's important to make the "...
7
votes
0answers
155 views

27 lines on a smooth cubic surface

It is known that every smooth cubic surface with coefficients in $\mathbb{Q}$ has $27$ lines defined over a number field extension of $\mathbb{Q}$ of degree at most $51840$ as the group $Gal(\overline{...
2
votes
1answer
137 views

The family of schemes $\operatorname{Spec} A[x]/(x^n)$

Consider the family $S_n:=\operatorname{Spec} A[x]/(x^n)$ of schemes, $A$ denoting any ring (which in our subject always means commutative and with identity). Is there some intuitive picture for $S_n$ ...
4
votes
1answer
78 views

Problem I.3.18 in Hartshorne

Problem I.3.18b-c in Hartshorne is concerned with the surface $Y$ of $\mathbb{P}^3$ given parametrically by $(x,y,z,w) = (t^4,t^3u,tu^3,u^4)$. In particular, part c asks to prove that $Y$ is ...
1
vote
1answer
46 views

The nonexistence of a polynomial

I'm studying algebraic geometry. To illustrate a nonalgebraic set, it is given that a unit circle except for a point on it in cartesian product or whole plane except for one point. Why doesn't a ...
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vote
0answers
78 views

“Implicit representations” of algebraic varieties

Consider a system of polynomial equations $S$ in multiple variables $x_1,\dots,x_n$ over the field $\mathbb{C}$. Is there a simple characterization of when the following property holds: There exists ...
2
votes
1answer
464 views

Finitely many singular points of an irreducible polynomial

let $k$ be a field, and consider an irreducible polynomial $f∈k[x,y]$. Let $S(f)$ denote the singular points of $f$ (points that are simultaneously zero on $f$, the $x$-derivative of $f$, and the $y$-...
0
votes
1answer
170 views

Veronese surface contains no lines

Why does Veronese surface contain no lines? Can you give me a reference about this fact? Thank you for your answers.
0
votes
1answer
64 views

Question regarding function field

I have learned in my algebraic curves class that the function field is the field of rational functions on a curve $C$ (or some variety). I was at a number theory talk, where the person counted the ...
1
vote
2answers
64 views

Enumerative projective geometry

I am wondering whether for any two lines $\mathfrak{L}, \mathfrak{L'}$ and any point $\mathfrak{P}$ in $\mathbf{P}^3$ there is a line having nonempty intersection with all of $\mathfrak{L}, \mathfrak{...
3
votes
1answer
306 views

Blow-ups in Projective Space

This is in regards to a question (no solutions or comments thus far :-() I asked earlier in regards to the blow-up of an elliptic curve: Question Let $f(x,y)=y^2-4x^3+ax+b$, where $(x,y)\in\mathbb C^...
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vote
2answers
53 views

The general expression of plane through the intersection of other two planes

For two planes: $$A_{1}x+B_{1}y+C_{1}z+D_{1}=0 $$ $$A_{2}x+B_{2}y+C_{2}z+D_{2}=0$$ Prove that any plane going through the intersection line of the previous planes could be expressed like where $\...
1
vote
1answer
118 views

Product of varieties in is a variety?

I know that the question may look similar to this: Is fibre product of varieties irreducible (integral)?, but I am forced by the context to use a different definition for variety. Definition. Let $K$ ...
2
votes
1answer
59 views

Standard proof that the set of sigularities is closed

I am trying to prove that the set of singular points of an affine variety is closed. Suppose $X\subset \mathbb{A}^n$. As every affine tangent space $T_x$ is embedded in this $\mathbb{A}^n$ we consider ...
1
vote
1answer
124 views

Number of points on an elliptic curve over $ \mathbb{F}_{q} $.

I have the following elliptic curve: $$ E: \quad Y^{2} = X^{3} + 1 ~ \text{over} ~ \mathbb{F}_{q}, ~ \text{where} ~ q \equiv 1 ~ (\text{mod} ~ 3). $$ I want to know the number of points on this curve. ...
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0answers
33 views

Some elementary questions on biprojective spaces

Suppose we define projective spaces over some field $k$, and consider the product $\mathbb{P}^{n_1} \times \mathbb{P}^{n_2}$. Unlike the affine case, we have $\mathbb{P}^{n_1} \times \mathbb{P}^{n_2} \...
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vote
2answers
142 views

Soft sheaves adapted to $f_!$

I'm reading Gelfand-Manin, Homological Algebra. I understand that the class of soft sheaves is sufficiently large, because every injective sheaf is soft. Now to see that this class is adapted to $f_!$,...
0
votes
1answer
77 views

Describing a tangent cone. What is that?

Could you please explain what a tangent cone is? For instance, consider the curve on $\mathbb{A}^2$ given by $f(x,y)=x^2-y^3=0$. Linear part is zero cause $\frac{\partial f(0)}{\partial x}=\frac{\...
4
votes
1answer
290 views

Question about Qing Liu's Algebraic Geometry book

I was just wondering what the real prerequisites are for reading Qing Liu's 'Algebraic Geometry and Arithmetic Curves', and if it is a good first book on the subject. In his preface he states that the ...
4
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0answers
100 views

Blowing up a Singular Point More Than Once.

I am trying to understand how $I_n$-fibres appear in an elliptic surface by performing a sequence of blow-ups. To be concrete, I am looking at the following elliptic surface given in Weierstrass ...
0
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2answers
53 views

Question regarding morphism of ringed spaces

I have recently started studying schemes, and I have encountered this passage from the book by Kenji Ueno: My questions: i) If $(X,O_X)$ is a local ringed space, why is $(X, i_*({O_X}_{|U}))$ also ...