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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7 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 ...
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0answers
10 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 ...
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1answer
18 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
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0answers
29 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 ...
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1answer
24 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 ...
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0answers
17 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 ...
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2answers
29 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 ...
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0answers
23 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 ...
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1answer
20 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 $f : Spec(k[x,y]) \to Spec(A)$ ...
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1answer
32 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 ...
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0answers
15 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 ...
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1answer
36 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 ...
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0answers
23 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, ...
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1answer
37 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 ...
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1answer
32 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. ...
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0answers
50 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 ...
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0answers
21 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 ...
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1answer
28 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 $I(Y)$? ...
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1answer
42 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 ...
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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 ...
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0answers
40 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 ...
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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 ...
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0answers
33 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
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1answer
40 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?). ...
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0answers
16 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 ...
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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 $Z \subset X$ ...
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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 ...
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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
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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 ...
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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 $\mathbf{A}^n$ ...
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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
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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, ...
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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
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1answer
37 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 ...
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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 ...
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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 ...
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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
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0answers
61 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 ...
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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 ...
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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 ...
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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 $C'\cap C'' = \emptyset$. ...
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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 : ...
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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 ...
5
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3answers
132 views

Basic application of the Nullstellensatz

Background: I have just started learning basic algebraic geometry. My solution to a simple problem involves an application of the Nullstellensatz and I want to know whether this is overkill (or ...
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1answer
35 views

Pushforward and sheaf-hom

If $S$ is a surface over the complex numbers $\mathbb{C}$, and $C$ is a curve in $S$, and $i:C\longrightarrow S$ is the inclusion morphism. If $A$ is a line bundle over $C$, then is it true that ...
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49 views

Is every reduced $k$-algebra all of whose residue fields are $k$ finitely generated?

Let $k$ be a field (of characteristic zero if you want). Let $A$ be a reduced $k$-algebra with the property that for every prime ideal $\mathfrak{p}$ of $A$ the natural homomorphism $k \to A/ ...
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32 views

morphisms of curves and discrete valuation rings

Given a dominant morphism $\varphi\colon C\to C'$ of curves, a nonsingular point $Q\in C'$, such that $\varphi^{-1}(Q) = \{P_1,\ldots, P_m\}$ consists of nonsingular points only. Then it is clear to ...
3
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1answer
74 views
+50

Proving that certain incidence correspondence is a projective variety.

Let $M$ be the projective space of nonzero $m\times n$ matrices up to scalars (in $\mathbb{K}$). In Joe Harris' Algebraic Geometry: A first course, in order to find the dimension of $M_{k}=\{A\in ...
3
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1answer
39 views

If $\dim(X)=\dim(Y)>0$, and $X\to Y$ is onto, does affineness of $X$ imply affineness of $Y$?

Suppose you have a pair of integral varieties $X$ and $Y$ such that $\dim(X)=\dim(Y)>0$, and there exists a surjective morphism $X\to Y$ between them. I was wondering, if $X$ is affine, does this ...
1
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1answer
42 views

Example of a divisor of a function

I'm studying Fulton's algebraic curves book and on page 97 Fulton defines the divisor of the rational function $z\in k(C)$: I'm looking for an example of a divisor like this one. Thanks