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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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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6 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
41 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
34 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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14 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
18 views

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
32 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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14 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
20 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
33 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
33 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
20 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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43 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
30 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
20 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
19 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
17 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
36 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 ...
2
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0answers
56 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
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1answer
35 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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34 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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37 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
25 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
12 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
126 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 ...
1
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1answer
32 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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48 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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22 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 ...
2
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1answer
63 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 ...
2
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1answer
37 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 ...
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53 views

Is this sheaf simple?

Let $S$ be a surface and $C$ be an effective divisor in $S$. That is $C$ is a curve in $S$ and $i:C\longrightarrow S$ is the inclusion morphism. Let $E$ be line bundle over $C$, so $i_*E$ is a ...
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1answer
40 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
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3answers
247 views

Geometrically, why do line bundles have inverses with respect to the tensor product?

Geometrically, why do line bundles have inverses with respect to the tensor product? Here my thoughts on the problem so far, please excuse their scatteredness. I know algebraically, it is just ...
0
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1answer
24 views

under change of coordinates the variety $Z(H_1,..,H_r)$ becomes $Z(x_1,…,x_r)\subset \mathbb{P}^n$.

A set $V\subset \mathbb{P}^n$ is called a linear subvariety of $\mathbb{P}^n$ if it's the zero locus of $r$ homogeneous and linear, i.e $V=Z(H_1,...,H_r )$ where each $H_i$ is a form of degree 1. I ...
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0answers
48 views

Meromorphic functions on $Y^2 = X^3 + 1$, genus.

Let $k$ be a field of characteristic $\neq 2$, and consider the quadratic extension $F$ of $k(X)$ generated by $\sqrt{X^3 + 1}$. What is/how do I find the genus of $F$? The progress I have so far: ...
3
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1answer
27 views

Exercise 1.13 of II in Hartshorne algebraic geometry.

The problem is following. 1.13 Espace Etale of a Presheaf. Given a presheaf $\mathscr F$ on $X$, we define a topological space $Spe(\mathscr F)$, called the espace etale of $\mathscr F$, as ...
5
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1answer
46 views

Only DVR's with quotient field $\mathbb{Q}$?

Let $p \in \mathbb{Z}$ be a prime number. I know how to show that $$\{r \in \mathbb{Q}: r = {a\over{b}},\text{ }a,b \in \mathbb{Z},\text{ }p\text{ doesn't divide }b\}$$ is a DVR with quotient field ...
3
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1answer
39 views

Book which covers these contents of the same level of Fulton's book.

My question is very specific. I'm studying Chapter 8 of Fulton's algebraic curves book and I would like to find another book (or online sources) which covers these contents: Divisors, the Vector ...
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1answer
35 views

Flat Module finitely generated when over the residue field finite dimensional? [on hold]

Let $(A, \mathfrak{m})$ be a local ring with residue field $\kappa=A/ \mathfrak{m}$. Let $M$ be a flat $A$-module. Assume that $M \otimes_A \kappa$ is a finite dimensional $\kappa$-vector space. Is it ...
5
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1answer
90 views

Complement of open set is finite in Zariski topology

This problem has two parts: a) Let $M$ be a finitely generated module over a Noetherian ring $A$. Prove that $S=\{ P \in\operatorname{Spec}(A) : M_P \mbox{ is a free }A_P\mbox{-module} \}$ is an ...
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0answers
40 views

Dimension of irreducible components of variety [on hold]

Consider the affine variety $X=\{ (a_1,a_2,a_3,b_1,b_2,b_3) \in \mathbb{C}^6 \mbox{ : }a_1b_2=a_2b_1, a_1b_3=a_3b_1 \}$. Prove that $X$ has two irreducible components, and that both of them are of ...
6
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1answer
49 views

Intersection of affine varieties is affine

Let $M,N\subset\mathbb{P}^n$ quasiprojective varieties such that there exist isomorphisms $i\colon M\rightarrow Z(a)\subset \mathbb{A}^m$ and $j\colon N\rightarrow Z(b)\subset \mathbb{A}^m$ for ideals ...
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0answers
17 views

Regarding Espace Etale of a presheaf, is $\bar s$ an open map?

I am reading Hartshorne Algebraic Geometry. In the exercise 1.13 of II on page 67, given a presheaf $\mathscr F $ on $X$ , $s \in \mathscr F (U)$, $\bar s : U\to Spe(\mathscr F)$ is defined by $P \to ...
2
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1answer
36 views

A Zariski open subset of a variety has the same dimension as the variety.

I am reading Joe Harris' book Algebraic Geometry: A first course. In the book he says: The Grassmannian $G(k,n)$ contains, as a Zariski open subset, affine space $\mathbb{A}^{k(n-k)}$, and thus ...
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0answers
23 views

Is there a $V$ such that $\operatorname{Pic}(V)\to\operatorname{Cl}(V)$ is not one-to-one?

Does there exist an example of an integral variety $V$ such that the usual map from the Picard group $\operatorname{Pic}(V)\to\operatorname{Cl}(V)$ is not one-to-one?
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3 views

Is any F-stable maximal torus contained in some F-stable maximal Borel subgroup?

Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and denote by $\bar{\mathbb{F}}_q$ its algebraic closure. Let $G$ be an affine algebraic group over $\bar{\mathbb{F}}_q$, and let $F$ be a ...
0
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0answers
20 views

Curves not contained in hypersurfaces

Consider a curve $C$ in $\mathbb{F}_q^m$, say. I am interested in the existence of curves not contained in any small degree hypersurface. For instance, a helix is not contained (or non-embeddable) in ...
2
votes
1answer
30 views

An injective morphism between varieties that is not an immersion

I believe this is relatively elementary, but I'm struggling to think of an example of a morphism $f: X \rightarrow Y$ between varieties which isn't an immersion in the sense of algebraic geometry. ...