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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about the spanned divisor of a complex variety

I have this definition: let $\xi \in H^1(X,O^*)$ a cocycle. We say that $\xi$ is spanned if for every point $x$ in my variety $X$ there exist a section $s \in H^0(X,O(\xi))$ such that $s(x) \neq 0$. ...
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15 views

are base change and restriction of scalars “inverses” in this case?

Let $l/k$ be a finite extension of fields. Let $G$ be an affine $k$-groups scheme. Let $G_l = G \times_k l$. Is it true that $\mathfrak{R}_{l/k}(G_l) = G$, where $\mathfrak{R}_{l/k}(-)$ is the ...
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14 views

When the fibers of a flat morphism are varieties.

Notations: As in Hatshorne's book. Suppose that $f:X\longrightarrow Y$ is a flat morphism between two non-singular projective varieties over an algebraically closed field. Are the fibers of $f$ ...
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22 views

Perimeter of an ellipse intuition help

I am aware that you can take the circumference of an ellipse using an elliptic integral and haven't looked much into it, but that seems to be a bit extreme and i was taking a personal look at conic ...
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13 views

$X\to \textrm{End}(O_{X,e}/m_{X,e}^r)$ is a morphism

Suppose $X^n$ is a complete group variety over algebaically closed field $k$, then the group law can be shown to be commutative. In proving this, one step is to show $X\to ...
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1answer
15 views

Intersection of open affines in a prevariety

What is an example of a prevariety in which the intersection of some two open affines is not an open affine? My examples of prevarieties that are not varieties does not extend beyond the affine line ...
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2answers
66 views

Etymology of “flabby” or “flasque” sheaf

I just started working with flasque, or flabby sheaves, that is sheaves whose restriction maps are surjective for any two open set of the space. I wonder about the etymology of the term. In French, ...
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2answers
32 views

Why aren't those Cartier Divisors equivalent?

Please refer to Gathmann's notes http://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/main.pdf at Example 9.3.6 for context. It's trying to give an example that the map between $Div(X)$ and ...
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0answers
28 views

Intersection of Segre variety with linear spaces

Consider the intersection of the Segre variety associated to product of $n$ copies of $\mathbb P^2$, with $k$ linearly independent hyperplanes. Is it possible to drop one of the hyperplanes and obtain ...
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1answer
19 views

Iitaka fibration over canonical model

I am looking for a referrence for the proof of following fact If the minimal projective manifold has positive Kodaira dimension and it is not of general type, it admits an Iitaka fibration over ...
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1answer
34 views

A particular example of a non-reduced scheme (with a reduced ring of global sections)?

I am searching for a scheme $X$ which can be obtained by gluing two affine schemes (along open subsets) such that: 1) X is non-reduced; 2) $\Gamma(X,\mathcal O_X)$ is a reduced ring. Any ideas? ...
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30 views

Quick question: Chern classes of Sym, Wedge, Hom, and Tensor

Given $L$ is a line bundle and $V$ is bundle of rank $r$ on a surface (compact complex manifold of dim 2). Recall the formula for $c_1$ and $c_2$: $c_1(V\otimes L)=c_1(V)+rc_1(L)$ ...
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13 views

About the holomorphic vector field tangent to a divisor

We say a holomorphic vector field $X$ is tangent to an effective divisor $D$, if $D_Xs=\lambda s$, where $s$ is the determining holomorphic section of the line bundle $L_D$ corresponding to $D$. If ...
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22 views

find points on circle in 3D pace perpendicular to line

I'm working with 3D image data and have little algebraic knowledge. I have an 3D image whit each pixel divined by its x,y,z position. What I need is to get the values of all pixels on a circle inside ...
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1answer
21 views

The vector bundle of an hypersurface

Suppose that $X$ is a compact complex variety and $V \subset X$ an irreducible hypersurface. Let $\{U_{\alpha}\}_{\alpha \in I}$ an open covering of X. With $s_{\alpha}$ i denote the local equation of ...
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15 views

Why is the function defined by $f(x_1,x_2)=0$ when $x_1=0$, and $x_2$ when $x_1\neq 0$ not regular?

I'm having trouble understanding what should be a straight forward example. Suppose $X\subseteq\mathbb{A}^2_k$ is cut out by the equation $x_1(x_2^2-x_1)=0$. Define a function $f:X\to k$ (here ...
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54 views

What happens when you drop “étale” from the construction of étale fundamental groups

Haven't thought about this deeply; it just came up at lunch and I wondered if anyone knew the answer. To define the étale fundamental group of a scheme $X$, we fix a point $x_0 \in X$, look at the ...
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35 views

Fibres of an ideal sheaf , total spaces and torsion groups

My question concerns a common example, which seems to often appear as an example/counter-example. Let $k$ be a field and consider the ideal exact sequence of the structure sheaf $k(p)$ of a point $p$ ...
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1answer
32 views

Zariski-open subset in $\mathbb{C}^n$ to Zariski-closed subset in $\mathbb{C}^{n+1}$

Let $n \in \mathbb{N}$, $f\in \mathbb{C}[X_1,\dots,X_n]$, and $D(f):=\{x=(x_1,\dots,x_n)\in \mathbb{C^n}|f(x)\neq 0\}.$ I want to show that there is a injective $\Phi: D(f) \rightarrow ...
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1answer
23 views

What is the $\pi_1$-action on the hom-sheaf between two finite etale covers?

Say you have two finite etale covers $X\rightarrow S$, $Y\rightarrow S$. The hom sheaf $\mathcal{H}om_S(X,Y)$ on the etale site $\text{Sch}/S$ is finite locally constant, hence representable by some ...
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2answers
63 views

Genus of intersection of two surfaces in $\mathbb{P}^3$

Let $F_1$ and $F_2$ be two (smooth) surfaces in $\mathbb{P}^3$, of degrees $d_1$ and $d_2$ respectively. Let $C$ denote curve given as their intersection. How one can compute arithmetical genus of the ...
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1answer
50 views

Is Spec ($k[x_1,x_2,\ldots])$ a smooth $k$-scheme?

Let $k$ be a field and let $A = k[x_1,x_2,\ldots]$. Note that $A$ is not of finite type over $k$. Is $\operatorname{Spec} A\to \operatorname{Spec} k$ a smooth morphism of schemes? I think it is ...
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1answer
52 views

Categories and the direct image functor

I've just started learning about sheaves and yesterday I had a look at MacLane/Moerdijk's book "Sheaves in Geometry and Logic". I've discovered something in this book on which I'd like some ...
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2answers
22 views

The interplay between projective and affine varieties.

I'm studying Algebraic Geometry first course from Harris and I didn't understand this equality: In another words, I'm having troubles to understand the interplay between $f_{\alpha}$ and ...
2
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1answer
45 views

generalized Euler exact sequence

I'm reading about Euler exact sequence in Ravi Vakil's notes, and I need help to check a few things. Given a scheme $X$, and a locally free sheaf $\mathcal{E}$ of rank $n+1$ on $X$, let us start from ...
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1answer
28 views

Fulton 8.17 ¿$\Gamma(X) = k$?

Problem 8.17 Let $X, Y$ be nonsingular projectives curves, $ f:X\to Y $ a dominating morphism. Prove that $ f (X)=Y $. Im trying to solve the problem 8.17 of Algebraic Curves Book of Fulton, there ...
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1answer
22 views

sheafification construction in Hartshorne

In section II.1 of Hartshorne, the sheaf $\mathscr F^+$ associated to a presheaf $\mathscr F$ is constructed so that $\mathscr F^+(U)$ is the set of functions $$ s\colon U \to \bigcup_{p \in U} ...
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1answer
42 views

The blow-up of $ X $

I'm studying blow-ups in connection with an introduction course in algebraic geometry. I've some problems with the details in the below set-up, which my textbook introduces in order to define the blow ...
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19 views

If the saturated ideal of hypersurface generated by one element?

If a subscheme of codimenion one in $\mathbb{P}^n_k$ is define by ideal sheaf $\mathcal{I}$, is the saturation $\oplus_{n\ge 0}\Gamma(I(n))\subseteq k[x_0,...,x_n]$ be generated by one element? Is ...
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1answer
34 views

Is the sum of saturated ideals saturated?

In a graded ring $S=\oplus_{k=0}^{\infty}S_k$, denote $m=\oplus_{k=1}^{\infty}S_k$, call an ideal $I$ to be saturated if $I=\cup_{n=1}^{\infty}(I\colon m^n)$. Is the sum of two saturated ideals still ...
2
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1answer
40 views

Rational functions on varieties

To give a rational function $f$ in the function field $K(X)$ of a smooth projective curve $X$ is equivalent to giving a finite morphism $f:X\to \mathbb P^1$. What is the precise analogue of this ...
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1answer
26 views

Rational functions over variety X

I 'm trying to solve this exercise of Fulton Algebraic Curves: Let $D=\sum n_P P$ be an effective divisor, $S=\{P \in X: n_P>0\}$, $U=X\setminus S$. Show that $L(rD)\subset ...
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1answer
65 views

What is $\mathrm{Proj}(S \otimes R)$?

What the title says. Let $S$ be a graded $k$-algebra, generated in degree $1$, and the same for $R$. Then $S \otimes_k R$ is graded as well, with $k$'th graded piece $\bigoplus_{j+l=k} S_j \otimes_k ...
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24 views

Definition of intersection multiplicity of a curve with some hyperplanes

I'm studying the chapter 2 of this paper and I have the following doubt: What is the definition of intersection multiplicity of a curve $C$ with some hyperplanes at a point $P$? Remark: My only ...
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28 views

Something like the Weierstrass preparation theorem?

I'm reading Griffiths and Harris, On the Noether-Lefschetz Theorem and Some Remarks on Codimension-two Cycles, Math. Ann. 271, 31-51 (1985). Their proof of the Noether-Lefschetz theorem is based on ...
2
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1answer
36 views

Confused about short exact sequence involving $\mathcal{O}_{\mathbb{P}^n}$ and $\mathcal{O}_Z$ for $\pi:Z\rightarrow \mathbb{P}^n$ closed embedding

If $\pi: Z\rightarrow \mathbb{P}^n$ is a closed embedding where $Z$ is the zero set of some degree homogeneous polynomial, we have: $$0\rightarrow \mathcal{O}_{\mathbb{P}^n}(-d)\rightarrow ...
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65 views

Why is such a the series algebraic but rational?

The coefficients of the series expansion of the algebraic function $A=\frac{1-\sqrt{1-8x^2}}{4x}$ are all intergers: $$A(x)=x+2x^3+8x^5+\cdots$$ But according to Polya's research,if $ F(x)$ is a ...
2
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1answer
39 views

First Chern class of line bundle corresponding to divisor

If I know an effective divisor $D$, then there is a line bundle $L_D$ corresponding to this divisor. How can I compute the first Chern class of $L_D$? For example, on $\mathbb{C}\mathbb{P}^3$, ...
2
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2answers
58 views

Stalks of the sheaf $\mathscr{H}om$?

The question is basically the title. What are the stalks of the sheaf $\mathscr{H}om_{\mathscr{O}_X}(\mathscr{F},\mathscr{G})$? If $X$ is a noetherian scheme and $\mathscr{F}$ is coherent, then ...
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1answer
36 views

Does a projective variety have a torus fixed point?

Let $X$ be a projective variety over $\mathbb{C}$ and let $T=(\mathbb{C^*})^k$ act on it. Is it true that there is a fixed point of this action on every irreducible component of $X$ just because $X$ ...
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37 views
+50

Pullback of principal Cartier divisors along a field extension

I tried the following problem in Liu's book, 7.3.1 but I don't see where it was needed that $X$ is integral - maybe someone can help me here. Is the following true without supposing that $X$ is ...
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1answer
35 views

Symmetry in complex plane

In a book I am reading, symmetry about a curve in complex plane is defined as follows: Let $F(x,y)=0$ be a simple curve. Then points $z, z_0$ are symmetric about this curve iff $ F \left( ...
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1answer
44 views

The ideal for image of Segre embedding

How to show the ideal $(X_{ij}X_{kl}-X_{il}X_{kj})_{0\le i,k\le m, 0\le j,l\le n}\subset k[X_{ij}]_{0\le i\le m, 0\le j\le n}$ is radical? I can show the zero locus defined by the ideal is the image ...
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1answer
27 views

If $K_X$ is not $\mathbb Q$-Cartier then it is not nef

Let $X$ be a projective variety. Is it true that if the canonical divisor $K_X$ is not $\mathbb Q$-Cartier then it is not nef?
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25 views

Is log-general type an intrinsic property of a variety

Let $X$ be a smooth quasi-projective variety over $\mathbb C$. Let us say that $X$ is of log-general type if for some choice of smooth compactification $\bar X$ with normal crossings boundary divisor ...
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39 views

Generalization of Bezout Theorem to many-hypersurface case in Hartshorne's setting

I try to follow the ideas in Hartshorne's Chapter 1, Section 7. Suppose we have algebraic sets $Y_1,...,Y_l$, I try to define their intersection number $I(Y_1,...,Y_l)$ to be the leading term of the ...
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1answer
28 views

Why is $V(x)\cup(\mathbb{A}^2\setminus V(y))$ not quasi-affine?

I'm having trouble understanding the following situation. Apparently it's not difficult to see the union $V(x)\cup(\mathbb{A}^2\setminus V(y))$ is not a quasi-affine set. Everything is being done ...
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2answers
36 views

Equation of a curved line that passes through 3 points?

I have a screen wherein the upper-leftmost part is at x,y coordinate (0,0). Then I have a curved line that passes through 3 points: (132, 201), (295, 661) and (644, 1085). Now, say I want to find 7 ...
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1answer
38 views

The first chern class of Fano manifold

If $M$ is a Fano manifold, $L$ is an ample line bundle over $M$. My question is that whether $c_1(L)=\alpha c_1(M)$ for some real number $\alpha$ always holds.
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16 views

Deforming unstable vector bundle to stable one

Let $V$ be a holomorphic vector bundle. If $V$ is strictly semistable, can we deform $V$ to a stable vector bundle, where stability is defined via the slope function?