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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Definition of homogeneous ideal

I'm a little confused about the definition of a homogeneous ideal. I have the following two definitions: An ideal $I\subset k[X_{0}, \dots, X_{n}]$ is homogeneous if $I$ is generated by (finitely ...
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1answer
40 views

Showing that an ideal is not generated by two elements

Let $X=V(x_1, x_2)$ and $Y=V(x_3, x_4)$ be affine varieties on $\Bbb C^4$ where $\Bbb C$ is the complex number. Then, I have to show that the ideal $I(X ∪ Y)$ cannot be generated by two elements. I ...
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30 views

Showing that stalk $O_{Spec \ A, p}$ is $A_p$

Suppose I have $A$, a commutative ring with unity. I would like to show that stalk $O_{Spec \ A, p}$ is $A_p$ for $p \in Spec \ A$. Could someone please explain me how this works? (I am having ...
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2answers
33 views

What are some good examples of (non-quasicoherent) sheaves not satisfying the conclusion of Hartshorne Lemma II.5.3?

Hartshorne, Algebraic Geometry, Lemma II.5.3 reads (roughly): Let $X = \operatorname{Spec} A$, let $f \in A$, and let $\mathscr{F}$ be a quasicoherent sheaf on $X$. (a) If $s \in \Gamma(X, ...
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Find length of the midpoints of the diagonals in a given trapezoid [on hold]

For any given trapezoid, where the bottom base, a, is larger than the top base b -- find the length of MN, the line connecting the midpoints of the diagonals, using only vectors.
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1answer
25 views

Analytically isomorphic fibers.

Suppose that $S$ is a non-singular complex projective surface with a fibration $f$ over $\mathbb P^1(\mathbb C)$. Suppose also that: There are only finitely many points $y_1,\ldots,y_n\in\mathbb ...
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27 views

Reference request: Galois descent

What is a classic (perhaps even original) reference for Galois descent? I know that it can be seen as a special case of faithfully flat descent (for which FGA and SGA I is the usual reference) and ...
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28 views
+50

When will the support of a non-effective Cartier divisor be pure of codimension 1?

Let $X$ be a scheme and $D \in Div(X)$ a non-effective Cartier divisor. I am curious as to when $\text{Supp } D$ is pure of codimension 1, i.e all irreducible components are of codimension 1. So, ...
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1answer
41 views

Irreducibility of holomorphic functions in a neighborhood of a point

Let $D \subset \mathbb C^n$ be a domain and let $f \in \mathscr O(D)$, $f \not\equiv 0$ be a holomorphic function. Define $$ V_f = \bigl\{ z \in D : f(z) = 0 \bigr\}. $$ Let $p \in V_f$. Suppose ...
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1answer
52 views

Hartshorne II prop 6.6

I'm having a really hard time understanding the proof of this proposition. $X$ is a noetherian integral separated scheme that is regular in codimension 1. We consider $X\times \mathbb{A}^1$ and the ...
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1answer
55 views

Riemann surface from $x^2 + y^2 = 1$ for $x,y \in \mathbb{C}$

I am reading Edward Frenkel's book Love and Math. In Chapter 9, it is talked about the one-to-one correspondence of solution of algebraic function of complex numbers and Riemann surfaces. can anyone ...
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0answers
19 views

A Question from the proof of affine algebraic group is linear

In the proof of the fact that any affine algebraic group is a linear algebraic group, there is an important step as follows (for example in Borel's book "Linear Algebraic Groups", Prop 1.10): Let $G$ ...
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0answers
26 views

Question regarding a section of an open set of the form $U \cup V$

Suppose I have some scheme $(X, O_X)$. Suppose I have two open subsets of X, $U$ and $V$. I was wondering about the following: 1) Is $\Gamma (U \cup V, O_X) \cong \Gamma (U , O_X) \times_{\Gamma (U ...
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1answer
36 views

homeomorphism over zariski topology becomes a homeomorphism over usual topology?

a semicubical parabola $L$ in $\mathbb C^2$ is given by $y^2=x^3$. I showed that a bijective function $f\colon\mathbb C \to L$ defined by $t \mapsto (t^2, t^3)$ becomes a homeomorphism regarding the ...
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0answers
26 views

Projective bundle is projective?

Let $\mathbb{P}(E)$ be a projective bundle over some smooth projective variety $X$, defined over $\mathbb{C}$ for definiteness. Then this bundle is also a smooth projective variety. Smoothness is ...
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1answer
25 views

What is the example of pseudo-effective divisor which is not an effective divisor

By definition, a pseudo-effective $\mathbb{R}$-divisor is the limit of effective $\mathbb{R}$-divisors in $N^1(X)$, I was wondering what is the example of pseudo-effective divisor which is not an ...
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1answer
44 views

A question about Klaus Hulek algebraic geometry

I'm reading Klaus Hulek's algebraic geoemtry and there is something that I can't understand. Here it says that if {p,q} is a counterexample with minimum max{deg p , deg q}, then it can be assumed ...
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1answer
22 views

Why is a projective subspace itself a projective variety?

Let K^(n+1) be a vector space and W be its subspace. Then projective subspace P(W) is said to be a projective variety of P(K^(n+1)). Could anyone tell me why it is so?
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27 views

semi-positiveness of canonical line bundle under the condition Kodaira dimension be positive.

Let $M$ be a projective variety with positive Kodaira dimension, then why the canonical line bundle is semi-positive?. Is there any reference?
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57 views

When is $A = k[x_1,\ldots, x_n]/I$ integrally closed?

Suppose that it is not easy to determine that $A$ is a UFD (or that it is a local, noetherian dimension 1 domain with principal maximal ideal). Can someone suggest strategies for showing that a ...
2
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1answer
33 views

Affine line with double origin

Let $X = Spec \ k[t]$ and $Y = Spec \ k[u]$ and let $U = D(t)$ and $V = D(u)$. I construct the affine line with double origin by gluing the two affine schemes $X$ and $Y$ together along $U \cong V$ ...
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1answer
37 views

Disjoint union of two affine schemes

Say I have two commutative rings with unity, $R$ and $S$. What does the sheaf of disjoint union of $\DeclareMathOperator{Spec}{Spec}(\Spec(R), \mathscr O_{\Spec(R)})$ and $(\Spec(S), \mathscr ...
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1answer
62 views

Analogue in algebra for characteristic classes?

By Swan's Theorem, we know that projective modules over a ring are an algebraic analogue of vector bundles over a base space. Is there some sort of cohomology theory of rings (or modules? or schemes, ...
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1answer
44 views

Structure sheaf of $Spec \ k[x,y]$

Let $k$ be a field. We consider the affine scheme $(Spec \ k[x,y], O_{Spec \ k[x,y]})$. Let $U = D(x) \cup D(y)$. We have that $\Gamma(D(x), O_{Spec \ k[x,y]}) = A_x$ and similarly $\Gamma(D(y), ...
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2answers
37 views

Derivative of a projective transformation

Assume $A$ is a matrix from $R^{n\times n}$, $A:R^n\rightarrow R^n$. Then $A$ induces a projective transformation $f:RP^{n-1}\rightarrow RP^{n-1}$. For example, $\\$ $$\begin{pmatrix} 4 & 0 ...
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92 views
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0answers
16 views

Quartic curves with four connected components

A quartic plane curve in $\mathbb{RP}^2$ can be defined by a quartic equation $F(x,y,z)=\sum a_{ijk}x^iy^jz^k$ with 15 coefficients. Now let's focus on smooth quartics that have a maximal number of ...
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1answer
19 views

Poles of functions defined on hyperelliptic curves

Consider the equation $y^2=P(x)$, where $P$ is a polynomial over a closed field $\mathbb{k}$ without multiple roots. Let $Y$ be the corresponding affine curve, $X$ - its nonsingular projective model. ...
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0answers
15 views

Suggestions for algebraic function fields papers

My professor of algebraic function fields class gave me a paper to make a project (give the proof details, fill some gaps, etc). As my previous question here suggests, the paper he gave me is hard for ...
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0answers
21 views

Base change is exact for algebraic groups

I need a reference for the following fact: let $1 \to G' \to G \to G'' \to 1$ be a ses of algebraic groups over $S$. Let $S' \to S$ be a base change. Then $1 \to G'_{S'} \to G_{S'} \to G''_{S'} \to ...
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1answer
53 views

Isogenies of abelian varieties

Let $\pi:X\to Y$ be a finite morphism of smooth projective curves over an algebraically closed field (of characteristic zero if necessary) which are both of genus $>1$. We have two "natural" maps ...
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2answers
43 views

Holomorphic Sphere $ S^2$ with (-1) self intersection number

What is the meaning of Holomorphic Sphere $ S^2$ with (-1)- self intersection number in intersection theory. Can we draw such sphere?
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1answer
46 views

Small deformations of smooth projective varieties

Let $X\subseteq \mathbf P^n(\mathbb C)$ be a non-singular projective variety over the complex numbers. Suppose that $X$ is given by the vanishing of homogeneous polynomials $f_1, \dots, f_r$. Is it ...
2
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1answer
46 views

Dense open subsets of schemes

Let $X$ be a scheme. Let $U$ be an open subset of $X$. It is clear that if $U$ contains all the generic points of $X$ (by which I mean the generic points of irreducible components of $X$) then $U$ is ...
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0answers
33 views

Cohomology and Base Change - Degree 0 Sanity Check

Both Vakil and Hartshorne describe Cohomology and Base Change in the following way: Suppose $f:X \rightarrow Y$ is a projective (in Vakil, proper) morphism of Noetherian schemes, $F$ a coherent ...
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0answers
20 views

Koszul complex and locally free resolution

Let $V$ be an $n$-dimenational vector space. We consider the tautological sequence on the Grassmannain $Gr_{k}(V)$ $$ 0 \to \Gamma \to V \times Gr_k(V) \to Q \to 0,$$ and the projection $p:V \times ...
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The riemann hypothesis [closed]

I'd like to speak the zeta functions that gives raise to the famous riemann hypothesis telling about the proprerties of its non trivial zeros. Last week i have finished my complex analysis course and ...
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0answers
48 views

From algebraic master degree to algebraic geomery Phd

I am a foreign master student in algebra at the final year. I'm familar with categorical algebra and have interesting in algebraic geomery and number theory. I have learned some knowledge about scheme ...
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1answer
27 views

Weierstrass points on algebraic curves

We are considering projective algebraic curves over a closed field $\mathbb{k}$. Let $X$ be such a curve, $\mathbb{k}(X)$ - the field of rational functions of $X$, $D$ - some divisor on $X$. We ...
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2answers
80 views

Is the Projective Real Plane Compact?

I feel like $\Bbb P (\Bbb R^2)$ is compact, but I know that $\Bbb R^2$ is locally compact, therefore it has a one-point compactification. $\Bbb P (\Bbb R^2)$ adds more than one point to the real ...
3
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1answer
49 views

Is a coherent locally free sheaf isomorphic it's dual?

Hartshorne chapter II problem 5.1 a) is to prove that the double dual of a coherent locally free sheaf $\mathscr{E}$ over a ringed space $(X,O_X)$ is isomorphic to $\mathscr{E}$. This can be done by ...
4
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1answer
81 views

Rational map of a curve to an elliptic curve

If I have a curve given by $$ y^2 = (x^3-1)(x^3-a), $$ how do I find out if there is a rational variable transformation $y=y(s,t)$, $x=x(s,t)$ that maps this curve onto an elliptic curve of the form ...
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54 views

A smooth cubic is not rational

We consider projective curves over the closed field $\mathbb{k}$. It can be proven that the curve is rational iff its genus $g=0$. Also the curve is birationally equivalent to a nonsigular cubic iff ...
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0answers
46 views

Lefschetz Hyperplane Theorem for Picard groups of surfaces?

Griffiths and Harris, On the Noether-Lefschetz Theorem and Some Remarks on Codimension-Two Cycles, Math. Ann. 271, 31-51 (1985), states [...] look at the restriction $$r_1 : ...
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25 views

Clarifying the definition of local equations of a subvariety

Let $X$ be an an algebraic variety over $\mathbb{k}$, $Y\subset X$ - its subvariety, $x\in Y$ - some point, $\mathcal{O}_x$ - its local ring. A family of functions $f_1,f_2,\ldots f_n$ is said to ...
2
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1answer
49 views

Varieties over a field $K$ are also varieties over any subfield of $K$.

Suppose that $f:X\longrightarrow\text{Spec} K$ is a variety over $K$, namely $X$ is an integral, separated $K$-scheme of finite type. Now if $L$ is a subfield of $K$, it is clear that there exists a ...
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0answers
21 views

Primality of homogeneous ideal

Let $R$ be the polynomial ring over the finite field $\mathbb{F}_p$ with $n$ variables. Let $I$ be an ideal of $R$ generated by homogeneous polynomials whose coefficients are 1 or -1. Are there any ...
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35 views

Isomorphism of stalks and the complement of the exceptional locus.

I am reading Qing Liu's book on Algebraic Geometry, and on pg. 272, the proof of lemma 2.20 b) there is a certain part I don't get. Let $X,Y$ be Noetherian integral schemes and let $f:X \rightarrow ...
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1answer
37 views

Under which assumptions counit of the adjunction $f^* f_* \to 1$ is epimorphic?

Let $f: X \to Y$ be a morphism of schemes. It produces a pair of adjoint functors $f^*$ and $f_*$ on the category of quasi-coherent sheaves i.e. there is a natural isomorpism $$ ...
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1answer
51 views

Quick question: a 2:1 map onto the projective line

Given a line $L$ in $\mathbb{P^2}$. How do we see that a surjective map $\mathcal{O}_\mathbb{P^2}^{\oplus2}\rightarrow j_{*}{\mathcal{O}_L(2)}$ ($j$ is the inclusion of $L$ to $\mathbb{P^2}$) ...