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. ...

learn more… | top users | synonyms (1)

0
votes
0answers
10 views

Birational equivalent and isomorphic representation of a subalgebra

here I am again with another exercise which gives me a hard time. Let $A$ be the subalgebra of $\mathbb{C}[t]$ of all polynomials $f(t)$ such that $f(1) = f(-1)$. Let X be an alebraic set such ...
2
votes
1answer
40 views

The elliptic curve $y^2 = x^3 + 2015x - 2015$ over $\mathbb{Q}$

Consider the elliptic curve \begin{equation*} E: y^2 = x^3 + 2015x - 2015~\text{over}~\mathbb{Q}. \end{equation*} I want to prove that $|E(\mathbb{F}_7)| = 12$, that $|E(\mathbb{F}_{19})| = 19$ and ...
0
votes
0answers
9 views

degree of meromorphic function bounded by given divisor

$X$ is a compact Riemann surface, with genus $g \geq 2$. $D$ is a divisor, $\deg(D) \geq 2g$. Now, for $f \in L(D)=\{\text{div}(f)+D \geq 0 \} $. What could we get about the map degree of $f$ as a ...
0
votes
0answers
19 views

Algebraic approach to Local Analytic Complex Geometry

I'm attending a second course in Complex Analysis from a geometrical point of view. In the final part of the course we have discussed about germs of analitic complex sets and their algebraic ...
1
vote
0answers
21 views

What is a projective rational function in Fulton's book?

I'm reading Fulton's algebraic curves and I don't know if I understood correctly this definition on the page 46: What does the author mean by $f,g\in \Gamma_h(V)$ being of the same degree? Since ...
0
votes
0answers
21 views

Fulton, algebraic curves exercise 4.11

I'm doing the exercises of the book Fulton Algebraic curves and I'm stucked in the following problem: A subset $V\subset\mathbb{P}^n(k)$ is a linear subvariety of $\mathbb{P}^n(k)$ if ...
1
vote
0answers
25 views

Subvarieties in projective spaces

I was doing the problems from Fulton of Algebraic Geometry, studying for my exam and I got stuck in this exercise. Describe all subvarieties in $\mathbb{P}^1$ and $\mathbb{P}^2$. Any help will be ...
1
vote
2answers
39 views

Compute the arithmetic genus of the union of the three coordinate axes $Z(x_1x_2,x_1x_3,x_2x_3) \subseteq \mathbb{P}^3$

Compute the arithmetic genus of the union of the three coordinate axes $Z(x_1x_2,x_1x_3,x_2x_3) \subseteq \mathbb{P}^3$ The arithmetic genus of $X \subseteq \mathbb{P}^n$ is ...
1
vote
0answers
19 views

Derived direct image of emdedding of projective varieties

Let $i:Y\to X$ be an embedding map between projective varieties. What is the example of $X$ and $Y$ such that the functor $Ri_*:D^b(\text{Coh}\,Y)\to D^b(\text{Coh}\,X)$ is not a fully faithful ...
2
votes
1answer
28 views

Why this set is the pole set of $z$?

Suppose $V\subset \mathbb P$ a projective variety, $z$ a rational function on $V$ and let's define $J_z=\{F\in k[X_1,\ldots, K_{n+1}\mid \overline Fz\in \Gamma_h(V)\}$ and $S_z$ the polo set of $z$, ...
2
votes
0answers
39 views

What's the most general geometry branch?

What is the most general geometry of curves and surfaces? For example, at curves, we define in differential geometry the tangent vector as the derivative of a regular curve, but visually many other ...
1
vote
1answer
29 views

Characterisation of closed subschemes of projective spaces

I'm trying to understand a step in the proof of corollary 5.16 in Hartshorne. The statement Let $ A $ be a ring. if $ Y$ is a closed subscheme of $\Bbb{P}_{A}^{r} $ then there is a homogeneous ...
0
votes
0answers
22 views

How to make 4 (parametrized) points (in the complex plane) concentric?

I consider a functional equation (see the earlier discussion at MO) $$ f(f(x))= f(x)^2 + x \qquad \text{where also} f(0)=0$$ In the following I write it in a more concise form (with $z$ instead of ...
0
votes
1answer
34 views

Describe the normalization of the cusp.

Show that the normalization of $A = k[x_1,x_2] / (x_2^2 - x_1^3)$ is isomorphic to $k[x]$ and describe (for $k$ algebraically closed) the induced map $Spec(k[x]) \to Spec(A)$ I know that $A$ is a non ...
0
votes
0answers
33 views

Elliptic curves and the reduction map

For $n \geq 1$ we write $E_n = \{(x:y:z) \in \ker(\pi)|x/y \in p^n \mathbb{Z}_p\}$ with $\pi: E(\mathbb{Q}_p) \rightarrow E(\mathbb{F}_p)$ the reduction map. I know that $\pi$ is a group morphism and ...
0
votes
1answer
27 views

Show that this map has not the going-down property.

Let $A= k[x_1,x_2,y] / (x_2^2-x_1^2(x_1+1))$ and $Spec(A) \to Spec(k[x_1,x_2,y])$ the natural inclusion induced by the projection $k[x_1,x_2,y] \to A$. Consider the map $f : Spec(k[x,y]) \to Spec(A)$ ...
1
vote
1answer
37 views

Endomorphisms in an exact sequence of vector bundles

Let X be a smooth projective variety over $\mathbb{C}$. And suppose we have an exact sequence of vector bundles over $X$. $\qquad\qquad\qquad\qquad\qquad 0\longrightarrow A\longrightarrow ...
3
votes
0answers
16 views

What is $HC_0(\operatorname{Spec} k[x,y]/(xy))$?

Does anybody know how to compute $HC_0(\operatorname{Spec} k[x,y]/(xy))$? Here $HC_0(-)$ is the zeroth cyclic homology group. I'm curious since $\operatorname{Spec} k[x,y]/(xy)$ can be viewed as the ...
0
votes
1answer
39 views

Intersection of ample and effective divisors

I believe it is something silly, but I'm a newbie, so why on a surface the intersection of an effective divisor and a divisor from ample bundle is non-negative? In fact, I need that an intersection of ...
0
votes
0answers
24 views

Making sense out of the definition for “morphism of geometric spaces”

I'm trying to read "Introduction to Algebraic Geometry and Algebraic Groups" by Demezure and Gabriel and I'm already stuck on the following definition. A geometric space is defined to be a pair $(X, ...
5
votes
1answer
40 views

Quick question: could someone clarify me the notion of “$k$-points” of a scheme?

Let $X$ be a scheme, and $k$ be a field. What does it mean by $X(k)$? The reason I am asking this is that there seems to be several definition(or convention) on it, and I can't see why they lead to ...
3
votes
1answer
34 views

$\overline{\mathbb{Q}}$-valued points of a $\mathbb{C}$-scheme

I've just read the definition of a $T$-valued point on an $S$-scheme from Ravi Vakil's notes (i.e. the morphisms $T \to S$), and something about the definition doesn't seem to make sense intuitively. ...
2
votes
0answers
52 views

Examples of vector bundles

I know three examples of interesting vector bundles (not taking into consideration what one gets from them using algebraic operations): Tangent bundle of a smooth manifold Tautological bundle over ...
0
votes
0answers
24 views

$H^1$ of some vector bundle on a cubic 3-fold

This question is a sequel to the following one Dimension of moduli space of some stable vector bundles on a cubic 3-fold. Let $E$ be a stable rank 2 vector bundle on a cubic 3-fold, say $X$, with ...
0
votes
1answer
29 views

Let $k$ be an algebraically closed field and $Y \subset \mathbf{A}^n$ be the set $\{(t,t^2,t^3|t \in k\}$. What is are the gnerators of $I(Y)$

I'm having a problem thinking through this rigorously. Let $k$ be an algebraically closed field and $Y \subset \mathbf{A}^n$ be the set $\{(t,t^2,t^3|t \in k\}$. What is are the generators of $I(Y)$? ...
2
votes
1answer
44 views

I need help in this proof of this exercise from Fulton's book

I'm reading Fulton's algebraic curves book. I'm trying to understand this solution which I found online of the question 4.17 on page 97. What I didn't understand is why $V(J_z)$ are exactly those ...
1
vote
1answer
45 views

If $Y$ is a quasi-affine variety, then dim$Y$ = dim$\overline{Y}$

Reading through the proof of proposition 1.10 in Hartshorne's Algebraic Geometry I found some of it to be unnecessary. Is the following proof correct or can you point out my flawed logic. Let $Z_0 ...
1
vote
0answers
41 views

Example 2.3.4 in Hartshorne algebraic geometry

I am reading Hartshorne algebraic geometry. Example 2.3.4. Let $k$ be an algebraically closed field, and consider the affine plane over $k$, defined as $A^2_k = \text {Spec} k[x,y]$ . The closed ...
1
vote
0answers
16 views

Classifying all $E$-torsors up to isomorphism?

If $0\to E\to F\xrightarrow{\phi}\mathbb{A}^1_X\to 0$ is an exact sequence of bundles over the scheme $X$, then it is known that if $s\colon X\to \mathbb{A}^1_X$ is the constant section sending ...
2
votes
0answers
41 views
+50

Classification of $3$-pointed rational curves

I tried to prove that $\mathbb P^1 \setminus \{0,1,\infty\}$ is the fine moduli space for the moduli problem, which assigns to a scheme $S$ the set of (isomorphim classes of) $4$-pointed rational ...
4
votes
1answer
41 views

Proof in Fulton's *Algebraic Curves*

I'm reading Fulton's algebraic curves book on page 106 and I didn't understand this proof: I didn't understand why can we assume $F_Y\neq 0$? (what $F$ irreducible has to do with this?). ...
1
vote
0answers
17 views

Equation of the curve corresponding to a principal polarization

Let $\mathbb{C}^2/\Lambda$ be a principally polarized abelian surface. I think it is well-known how to write down the equation of the divisor (Riemann surface) corresponding to the polarization, in ...
1
vote
0answers
21 views

Nakai-Moishezon Criterion for effective $k$-cycles instead of only integral subschemes

The Nakai-Moishezon Criterion states that a Cartier divisor $L$ on a proper scheme over a field is ample if and only if $L^{\dim(Z)} \cdot Z > 0$, for every closed integral subscheme $Z \subset X$ ...
0
votes
1answer
40 views

Showing that $\mathbb{P}_k^2$ is birational to $\mathbb{P}_k^1 \times \mathbb{P}_k^1$

I wanted to check that there was nothing (roughly) wrong with my reasoning in showing that $\mathbb{P}_k^2$ is birational to $\mathbb{P}_k^1 \times \mathbb{P}_k^1$. First of all, I know that for two ...
1
vote
1answer
35 views

Help to understand this proof in Fulton's book

I'm reading Fulton's algebraic curves book on page 105 and I didn't understand this proof: 1.Why if $R=k[X_1,\ldots,X_n]$, then $\Omega_k(R)$ is generated (as R-módule) by the differentials ...
2
votes
1answer
21 views

If $V$ irreducible an. variety $V=V_1\cup V_2$, then $V_1\cap V_2\subset V_{sing}$

Let $V$ be an irreducible analytic variety, and $V_1, V_2$ analytic subvarieties such that $$V=V_1\cup V_2.$$ In Griffiths-Harris book, it is mentioned that $V_1 \cap V_2$ is a subset of the ...
0
votes
0answers
46 views

Help with Proposition 1.13 in Hartshorne's Algebraic Geometry

This is Proposition 1.13 in Hartshorne's Algebraic Geometry. I just need to make sure that the following proof of one direction is correct. I am trying to show that if a variety $Y$ in $\mathbf{A}^n$ ...
1
vote
1answer
24 views

How do I find the orders of this rational function?

How can I find the orders of $z(x)=\frac{x}{1-x}$ over $k(\mathbb P^1)$ at the zero $x=0$ and the pole $x=1$? I saw in another question posted on MSE that the orders are both equal to $1$, but I ...
2
votes
1answer
32 views

definition of singular locus of a variety

Given a variety $X$ over $k$, we can consider which points are regular, and we can define the singular locus $\operatorname{Sing}(X)$ as the complement of the regular points in $X$. My question is, ...
2
votes
0answers
22 views

Irreducibility of analytic varieties

Let $V$ be an analytic variety and $V^{*}$ denote the locus of its smooth points. From Griffiths & Harris, page 21, we have that an analytic variety $V$ is irreducible iff $V^{*}$ is connected. ...
2
votes
1answer
38 views

Zeroes of prime polynomials in the algebraic torus (A Hilbert's Nullstellensatz for Laurent polynomials?)

Let $Q\in\mathbb C[z_1,\dots,z_D]$ be a prime polynomial and let $Z(Q)$ be the algebraic hypersurface of its zeroes. Assume that $P\in\mathbb C[z_1,\dots,z_D]$ is a polynomial which has zeroes at ...
0
votes
0answers
21 views

Describing locally the Fano variety (Harris: First Course, Example 6.19)

Let $G$ be a homogeneous polynomial of degree $d$ in $n$ variables over an algebraically closed field $K$. Let $F_s(X)$ be the set of all $s$-dimensional linear subspaces of $K^n$ that are contained ...
1
vote
0answers
18 views

Singular locus of analytic subvarieties

In Griffiths and Harris page 21, it is proven that the singular locus, denoted $V_{s}$ is contained in an analytic subvariety of the complex manifold $M$ not equal to $V$ which is the analytic ...
0
votes
1answer
38 views

Help me with this solution of the exercise 4.17 from Fulton's Algebraic Curves

I'm studying Fulton's algebraic curves book and in order to prove the well-definiteness of the divisor $div(z)$ on page 97 I'm trying to understand this solution which I found online. I didn't ...
3
votes
0answers
62 views

Zariski density of points over completion

I have a simple question which I couldn't find a reference to. Let $X$ be a smooth projective irreducible variety over $\mathbb{Q}$. Suppose we base change to $\mathbb{Q}_p$ (the $p$-adics) and ...
2
votes
1answer
36 views

Finite covering by finitely generated B-algebras

While I was working on the first properties of schemes on Hartshorne's Books, I needed the following result that I couldn't prove it: Let $X=Spec(A)$ be an affine scheme. Suppose that there exist an ...
-1
votes
0answers
37 views

Endomorphism of projective bundle

Let $X$ be smooth projective variety. Let $E$ be a vector bundle over $X$. Is $End(E)=End(\mathbb{P}(E))$? Given a $\phi:E\longrightarrow E$, we get a morphism ...
0
votes
0answers
44 views

Hartshorne Exercise V.2.2

This exercise is to prove a ruled surface $X=\mathbf{P}(\mathscr E)$ over a curve $C$ is decomposable if and only if there exist two sections $C'$ and $C''$ of $X$ such that $C'\cap C'' = \emptyset$. ...
0
votes
0answers
26 views

Bertini's papers in english

I'd very interested by finding translation in english of some papers of Bertini. I don't want necessarly a paper from himself, for exampe I was very happy to find this paper : ...
0
votes
0answers
13 views

Dirrefential of boundary morphisms in the moduli space of pointed stable curves.

Recall that first order deformations of a smooth pointed curve $(C,p_1,\ldots,p_n)$ are parametrized by $H^1(C,\cal{T}_C(-p_1-\ldots-p_n))$ and in the stable case is ...