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

learn more… | top users | synonyms (1)

1
vote
2answers
55 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
56 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 ...
1
vote
0answers
61 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
0answers
57 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 $m_A$, ...
6
votes
1answer
58 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
44 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 ...
1
vote
2answers
177 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 ...
1
vote
0answers
52 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 $v$ ...
11
votes
2answers
404 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 ...
9
votes
0answers
115 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 ...
0
votes
0answers
50 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
33 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
31 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
0answers
27 views

definition of semidirect product on two projective spaces

What is the definition of semidirect product on two projective spaces $\mathbb CP^1 \rtimes\mathbb CP^2$ I can not undrestand it.
0
votes
1answer
34 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
44 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.
0
votes
1answer
73 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
74 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 ...
5
votes
0answers
74 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 ...
2
votes
1answer
88 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 ...
4
votes
1answer
52 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
43 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 ...
0
votes
0answers
58 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 ...
1
vote
1answer
54 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 ...
0
votes
1answer
62 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
0answers
32 views

Finding the point satisfying the condition

Given N interesting points on the plane. Each interesting point has integer coordinates. Also, all the interesting points form a strictly convex polygon. If we select two coordinates from these ...
0
votes
1answer
35 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
36 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}, ...
3
votes
1answer
107 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 ...
2
votes
2answers
38 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 ...
2
votes
1answer
36 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
46 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
71 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. ...
1
vote
0answers
21 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} ...
1
vote
2answers
92 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 ...
0
votes
1answer
35 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 ...
2
votes
1answer
68 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 ...
1
vote
0answers
33 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
votes
2answers
47 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 ...
0
votes
1answer
40 views

Tangent bundle for the projective plane curve

Consider the cubic $C$ with an equation $x_0^3+x_1^3+x_2^3=0$ (this is a projective curve on $\mathbb{P}_2=\mathbb{P}(V)$). I need to find the equation of the closure of all tangents to $C$ (it is ...
1
vote
0answers
18 views

Rank of derivative polynomial map equals dimension image?

I've been told that given a polynomial map $f:X\to Y$ in characteristic zero, there exists an open dense subset $U$ of $X$ such that for all points $x$ in $U$, the rank of the derivative of $f$ in $x$ ...
2
votes
0answers
108 views

Locally trivial morphism into the Jacobian

Let $C$ be a smooth projective curve of genus $g$ over a field $k$ and let $J$ denote its Jacobian. Let $P$ be a $k$-rational point on $C$, and let $r$ be a natural number. Then there is a morphism ...
0
votes
1answer
49 views

What is a Presheaf (intuitively) and help with the technical machinery.

I have come across things such as that a Presheaf $\mathcal{F}$ associates data (such as rings, groups, other sets etc.) to open sets $U$ of $X$. That the Presheaf $\mathcal{F}$ becomes a Sheaf if ...
1
vote
1answer
58 views

3 points collide in $\mathbb{C}^2$

In Nakajima's book, "Lectures on Hilbert Schemes of Points on Surfaces", he gives an explicit description of the corresponding ideal for two points colliding in $\mathbb{C}^2$. This basically ...
5
votes
1answer
59 views

From a vector bundle to a Koszul complex

Let $k = \mathbb C$. Given a commutative $k$-algebra $A$, an $A$-module $M$ and a homomorphism of $A$-modules $s:M \to A$, we can construct the Koszul dg algebra. $$K(A,M,s) = \wedge^{-\!*}_A(M)$$ ...
0
votes
1answer
64 views

Prove: $U \mapsto \mathrm{Hom}(U, Y)$

Rewording this problem via what Zhen Lin's notion of the original question is. For $X$ and $Y$ ringed spaces Prove: For each open $U \subset X$ the Presheaf $U \mapsto \mathrm{Hom}(U, Y)$ is a ...
1
vote
0answers
37 views

Variety $V=\{(x,y)\in k^2\mid xy=1\}$ is connected

Let $k$ be an infinite perfect field. Show that the variety $$V=\{(x,y)\in k^2\mid xy=1\}$$ is connected of dimension $1$. Many thanks in advance.
3
votes
3answers
100 views

How to imagine “tensoring with Serre's twisted sheaf”

What has an algebraic geometer in mind when (s)he sees $\otimes \mathcal{O}(1)$? I think it has something to do with an intersection of a hypersurface...? Thanks, Adrian
1
vote
2answers
60 views

Equation $1+x^8y^4+x^4y^8-x^2y^4-x^6y^6-x^4y^2=0$

How to prove that the following equation: $$1+x^8y^4+x^4y^8-x^2y^4-x^6y^6-x^4y^2=0$$ has for solution(in real numbers): $|x|=|y|=1~$ only. Any hint would be appreciated.
3
votes
1answer
38 views

Degree of ample bundle over projective curve is positive

(From Vakil's notes, Exercise 18.4.K) If $C$ is an integral projective curve over a field $k$, and $\mathscr{L}$ is an ample line bundle on $C$, why is the degree of $\mathscr{L}>0$? If $C$ is ...