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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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
14 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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13 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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43 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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20 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
21 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
56 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
49 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
49 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 ...
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1answer
43 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
27 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
19 views

Graded ring localization. Why is this function bijective?

From Hartshorne, chapter II.2, proposition 2.5-b. If R is a graded ring and a is a homogenous ideal, then the function defined as $\phi(a) = (a.R_f)\cap R_{(f)}$ is a bijection. Where $R_f$ is ...
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1answer
21 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
41 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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18 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
32 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
36 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
53 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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23 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 ...
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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
35 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
33 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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36 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
25 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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15 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?
3
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1answer
44 views

Layman's Question on Schemes

I am reading Jordan Ellenberg's article on Arithmetic Geometry in the Princeton Companion to Mathematics. I have forgotten most of the algebra I learned since passing my qualifying exams more than 30 ...
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1answer
25 views

How to visualize d-Uple embedding?

This may be a vague question and please feel free to edit it. Is there any good way to image what d-Uple embedding looks like? Thanks!
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8answers
103 views

How to show $P^1\times P^1$ (as projective variety by Segre embedding)is not isomorphic to $P^2$?

I am a biginner. This is an excise from Hartshorne Ch 1, 4.5. By his hint, it seems this can be argued that there are two curves in image of Segre embedding that do not intersect with each other ...
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0answers
63 views

The Set of All Integers is NOT a Variety; How Come?

My understanding is that a variety is, essentially, a set of common "zeros" of some given functions in the given ring. My professor told us that a finite set of integers form a variety; however, the ...
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1answer
34 views

Irreducible components of an Algebraic subset.

This is question 1.27 from Fulton's textbook: http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf (the very top of page 9). 1.27. Let $V, W$ be algebraic sets in $\mathbb{A}^n(k)$, with $V\subset ...
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45 views

Algebraic Geometry: A question about radical ideal

I'm working on this problem: Show that a rational normal curve $C$ of degree $3$ cannot be an intersection of two quadrics. Here is my solution. Let $$J=\{\text{$f$ is homogeneous of degree $2$ and ...
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27 views

birational map dominant?

I've run across a theorem stating that a birational map f : w -> w is dominant. birational means there is another rational map g : w -> v such that the compositions of f and g are identity maps. ...
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40 views

Is there a plausible outline of how geometric complexity theory could prove $P \neq NP$?

I've heard people saying that geometric complexity theory could be the key to showing $P \neq NP$, but when I've actually read about it it seems like it's concerned with other, perhaps analogous ...
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64 views

three axis in $\mathbb{A}^3$ can't be defined by two functions

I am reading Shafarevich's book on Algebraic Geometry and in 1.6.5, exercise 3. He asks to prove that $X \subset \mathbb{A}^3$, which is the union of the three coordinate axis, can not be defined by ...
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1answer
55 views

Prove that $S$ is an integral domain and $T$ is not an integral domain.

Let $R = \mathbb{C}[x,y]$ $R^i \subset R$ be the abelian subgroup of $R$ generated by elements of $\mathbb{C}$ times monomials of degree at least $i$ $I = (x^3+x^2-y^2)$ $S = R/I$ $S^i$ be the group ...
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1answer
30 views

Finding a line $L\subset V(y-xz)\subset\mathbb A^3_k$

I want to find lines $L\subset V(y-xz)$ and $M\subset\mathbb A_k^2$ such that $$ V(y-xz)\setminus L \simeq \mathbb A_k^2\setminus M\ . $$ Hint suggests that I use the projection $(x,y,z)\mapsto(x,y)$. ...
3
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1answer
72 views

Question on how to get back “classical” Serre-duality from its derived functor formulation

I'm really new to derived categories, so i hope this isn't a stupid question. I'm trying to understand how the duality described as for example in Residues and Duality of R. Hartshorne, using the ...
5
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0answers
53 views

Basic question: Condition for a map associated to a linear series to be an immersion

I am reading this set of lectures of a class by Prof. Harris. There is a theorem. Let $X$ be a Riemann surface and $\phi:X\rightarrow\mathbb{P^r}$ be the map defined by a linear series without ...