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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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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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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19 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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13 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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10 views

Isotrivial, elliptic and semistable fibrations over $\mathbb P^1(\mathbb C)$

I need to know if it is possible to construct a smooth complex projective surface $S$ with a relatively minimal fibration over $\mathbb P^1(\mathbb C)$ which has the following properties: The ...
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19 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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34 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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16 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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24 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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24 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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33 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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21 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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26 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
34 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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64 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
32 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 $\mathbb{C}\mathbb{P}^3$, ...
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2answers
55 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
30 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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31 views

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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43 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
23 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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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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38 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
27 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
35 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
37 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?
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1answer
43 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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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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62 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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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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39 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
52 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 $(x,y,z)\mapsto(x,y)$. ...
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1answer
70 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 ...
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51 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 ...
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23 views

Determine sine wave frequency from two arbitrary points

If I have only two arbitrary points on a sine wave, what would be the simplest method for determining the frequency of the sine wave? The frequency is unknown. The bandwidth is restricted, the time ...
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rational normal curve of degree 3 not written by intersection of two quadrics

I'm learning about rational normal curves of degree n. And the book says that rational normal curves of degree 3 cannot be written by intersection of two quadrics. I can visualize the situation in my ...
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1answer
37 views

Showing local ring isomorphisms

This is a problem in K. Hulek's Elementary Algebraic Geometry. I figured out that $k[X]$ is the collection of polynomials of the form $f(x) + g(y)$ and also the local ring of an affine line at the ...
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43 views
+100

Number of fibrations over a curve.

Fix a non-singular complex projective curve $C$. I would like to know how many non-singular complex projective surfaces $S$ have the following properties (up to isomorphism): There is a fibration ...
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1answer
17 views

Are quasiaffine subsets of $\mathbb{A}_F^n$ always necessarily open or closed?

Something I was wondering about lately, suppose $\mathbb{A}_F^n$ is affine space over a field $F$ which is algebraically closed. Are the quasiaffine subsets $Z\subseteq\mathbb{A}_F^n$ always either ...
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1answer
64 views

To show a morphism of affine k-varieties which is surjective on closed points is surjective

This is a exercise from Ravi Vakil's Foundations of Algebraic Geometry, Ex 7.4.E. Assume Chevalley's theorem. Show that a morphism of affine $k$-varieties $\pi:X \rightarrow Y$ is surjective iff ...
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30 views

Showing that the image of a polynomial map is not closed

Let $f : \mathbb{C}^3 \rightarrow \mathbb{C}^4$ be defined by $(s, t, u) \rightarrow (st, st^2+(1-s)u, st^3, 1-s)$, where $\mathbb{C}$ denotes the complex numbers. Then for some irreducible ...
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1answer
36 views

Strict Transform of a Line in a Blow Up

Consider the blow up $\pi:B \to \mathbb{A}^2$ of the origin in $\mathbb{A}^2$. Let $L=Z(ax+by)$ be a line through the origin in $\mathbb{A}^2$ and let $\widetilde{L}$ be the strict transform of $L$ ...
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+50

Pre-requisites and references for $K3$ surfaces

I would like to know the "roadmap" to study $K3$ surfaces. Perhaps, my background might be helpful: I am an undergraduate student, who knows the basics of Differential Geometry, Topology, Complex ...