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)

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
69 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
34 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
60 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 ...
0
votes
0answers
38 views

Unstable extension rank two vector bundle

I'd like to classify the not semistable extensions of this form: $$0 \to O_C \xrightarrow{i} E \xrightarrow{\pi} L \to 0$$ where $C$ is a curve and $L$ is a line bundle isomorphic to the determinant ...
1
vote
1answer
142 views

Proof of rigidity lemma

I am trying to understand in full details the proof of rigidity lemma as proved here http://staff.science.uva.nl/~bmoonen/boek/DefBasEx.pdf [Lemma 1.11, Pag. 12]. The statement in this reference is ...
0
votes
1answer
83 views

Studying $\operatorname{Spec}\mathbb{Z}[x]$, $\operatorname{Spec}\mathbb{R}[x]$, and $\operatorname{Spec}\mathbb{C}[x,y]$.

While there is a similar question here but that was marked as a duplicate to this question. The latter question, at the level that I am at doesn't give me much insight. I also thought that if I could ...
2
votes
1answer
38 views

A condition for a homogeneous ideal to be prime

The following is the problem 11 of Chaper 8 Section 4 of Ideals, Varieties, and Algorithms by Cox, Little and O'Shea. A homogeneous ideal is said to be prime if it is prime as an ideal in ...
0
votes
1answer
32 views

What is the intuition behind contours and their geometric properties

What is the the intuition behind contours? Can someone explain whar are contours, their geometric properties in simple manner
2
votes
1answer
169 views

Resolution of Singularities: Base Point

Consider the curve $y^2=4x^3-ax-b$, where $a$ is a fixed constant and $b$ is a free constant. For each value of $b$ we get a family of curves. Part 1: Show that the family of curves intersect at ...
2
votes
1answer
54 views

cotangent space

Let $K$ be a field and $L$ be its field extension, $K\subset L.$ Let $V \subset L^n$ and let $\mathcal{I}(V)\subset K[x_{1},...,x_{n}]$ be its vanishing ideal. Let ...
0
votes
1answer
46 views

Noether normalization and surjectivity (revisited)

Let $Y$ be an affine variety of dimension $d$ inside the affine space $\mathbb{A}^n$. Then $A(Y) = k[x_1,\dots,x_n]/I_Y=:k[\bar{x}_1,\dots,\bar{x}_n]$. By the Noether normalization theorem, there ...
1
vote
1answer
47 views

a question on higher direct images in a product

I want to compute $H^i(X,\pi_1^* O(a)\otimes \pi_2^*O(b))$ where $X=\mathbb{P}^1\times \mathbb{P}^1$ and $\pi_i$ is the projection onto the $i$-th $\mathbb{P}^1$ factor. I realize this is a special ...
1
vote
1answer
55 views

About globally generated Sheaves

On Vakil's Lecture Notes, he puts an important exercise that says: ''Suppose $\cal{F}$ is a finite type quasicoherent sheaf on a scheme $X$. Show that$\cal{F}$ is globally generated at $p$ if and ...
0
votes
0answers
23 views

The closure of semialgebraic sets is semialgebraic.

I want to prove that the closure of semialgebraic subsets of $\mathbb{R}^n$ with respect to the Euclidean topology is semialgebraic. I may use the Tarski–Seidenberg theorem. Please give me not the ...
1
vote
0answers
40 views

global section of some sheaves

Let $\mathrm{Grass}(r,V)$ be the Grassmannian over a field $k$. What is $H^0(\Sigma^{\alpha}(S))$, where $S$ is the tautological sheaf and $\Sigma$ the Schur functor. In characteristic zero this is ...
1
vote
1answer
80 views

pullback of canonical divisor

Let $Y$ be a smooth variety of dimension $n$. Then I can get (a representative for) the canonical divisor class $K_Y$ on $Y$ by taking any rational $n$-form $\omega$ on $Y$ and taking its divisor of ...
2
votes
2answers
76 views

sections of an invertible sheaf, and their support

Suppose I have a section $s$ of an invertible sheaf $L$, vanishing along a divisor $D$. Then there is an isomorphism $(L, s) \simeq (O(D), 1)$. In the next paragraph I'll pick $D=K_X$, but that is ...
-1
votes
2answers
37 views

Definition of quasi-projective variety and some related questions

I'm a little bit confused by Definition 1.64 on page 32 in this book. This definition says: A prevariety is called quasi-projective variety if it is isomorphic to an open subvariety of a projective ...
1
vote
1answer
95 views

Hartshorne “Algebraic Geometry” theorem 8.15

The theorem $8.15$(p.177) from the Hartshorne's book "Algebraic Geometry" says: " Let $X$ be an irreducible separated scheme of finite type over an algebraically closed field $k$. Then ...
0
votes
1answer
44 views

Grade of non principal Prime ideals in Noetherian UFDs

I want to prove that in any Noetherian UFD the grade of every non-principal prime ideal is at least $2$. I say in a UFD $R$ each nonzero prime ideal contains a prime element. Since the given ...
0
votes
0answers
78 views

Ex 6.7 in Görtz and Wedhorn's AGI

From Algebraic Geometry I, by Ulrich Görtz and Torsten Wedhon, p.165: Exercise $\boldsymbol{6.7}$. Let $k$ be a field, let $X$ be a $k$-scheme, and let $Y_1$ and $Y_2$ be closed subschemes of of ...
0
votes
0answers
30 views

Criterion of separability of function on a curve.

Let $K$ be finitely generated extension of an algebraically closed field $k$ of transcendence degree $1$, $\operatorname{char }k =p$. Let $C(K)$ be set of discrete valuation rings $(\mathcal ...
1
vote
1answer
47 views

tensor product of presheaves of modules

Let $\mathscr{O}$ be a presheaf of rings on $X$ and $\mathscr{F}$, $\mathscr{G}$ be presheaves of $\mathscr{O}$-modules on $X$. Let $\mathscr{O}^{\#}$,$\mathscr{F}^{\#}$ and $\mathscr{G}^{\#}$ be ...
1
vote
1answer
43 views

Twisting relative proj (exercise from Vakil)

I'm stuck on another problem (17.2.G) from Vakil's notes, and I'm wondering if somebody could get me started. Specifically, we are given a scheme $X$, an invertible sheaf $\mathscr{L}$ on $X$ and a ...
0
votes
1answer
41 views

Simple questions about projective space

I have a few questions concerning page 28 of this book. Q1: In the definition of $\mathcal{O}_{\mathbb{P}^n(k)}(U)$ we have to check if $f_{\mid U\cap U_i}\in \mathcal{O}_{U_i}(U\cap U_i)$. We did ...
1
vote
1answer
56 views

Ring extension and Jacobson rings

If $R\subseteq S$ are commutative rings, is it a fact that $R$ is a Jacobson ring if and only if $S$ is so? I guess the contraction of maximal and prime ideals of $S$ may be helpful in this ...
1
vote
1answer
67 views

Beggining in Algebraic Geometry

My question is about sources for start the study of algebraic geometry. I know that it requieres so much algebra, but, is there any book which can be readed without many tolos of modules, Galois, ...
0
votes
1answer
51 views

Projective variety minus hyperplane $=$ affine variety

Claim: Let $V \subset \mathbb{C}P^n$ be a non-singular projective algebraic variety of complex dimension $k$ and let $P \subset \mathbb{C}P^n$ be a hyperplane. Then $V \setminus (V \cap P)$ is a ...
0
votes
1answer
43 views

Nullstellensatz non-valid for non-algebraically closed fields

I want an example (with details, please) showing that Nullstellensatz may be false over non-algebraically closed fields. Thanks in advance!
0
votes
0answers
43 views

About the cohomology of a projective noetherian scheme

In Hartshorne's book, we have in the theorem III, 5.2 : Let $X$ be a projective scheme over a Noetherian ring, and let $\mathcal{O}_{X}(1)$ be a very ample invertible sheaf on $X$ over $SpecA$. Let ...
1
vote
1answer
61 views

what are the “coordinates” of an affine variety?

Let $X$ be an affine variety of dimension $d$ inside the affine space $\mathbb{A}^n$. What do we mean when we say "let $u_1,\dots,u_d$ be coordinates of $X$"? Does this mean that we can describe every ...
1
vote
1answer
38 views

curves $C$ in surfaces with $C^2<0$ and $C$ is not rational

What is an example of a smooth surface $X$ with an irreducible curve $C \subseteq X$ with $C^2<0$ but $C$ is not rational? I don't know how to make curves with negative self intersection other than ...
2
votes
0answers
31 views

Infinitesimal deformation of coherent sheaves

Let $\mathcal F$ a coherent sheaf over an affine subset U, then we can consider it as an R module, if $U=Spec(R)$. Let $R$ an algebra over an algebraically closed field $\mathbb K$ and consider the ...
3
votes
1answer
72 views

Application of Riemann Roch

I have read that thanks to Riemann Roch theorem, if get $\Sigma$ a compact Riemann Surface of genus $g$ there exists a conformal branch covering $\phi: \Sigma \rightarrow S^2$ of degree less than ...