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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1answer
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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( ...
2
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1answer
45 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?
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1answer
45 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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108 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
64 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
35 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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0answers
46 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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0answers
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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0answers
42 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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0answers
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
56 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
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 ...
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0answers
56 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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0answers
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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0answers
26 views

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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0answers
62 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
69 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 ...
2
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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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0answers
61 views
+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 ...
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1answer
40 views

automorphism of the projective space $\mathbb{P}_A^n$

In exercise 16.4.B of Vakil's notes, he establishes that the group of automorphisms of $\mathbb{P}_k^n$ is $PGL_{n+1}(k)$. This I can manage to show, but in the remarks following the exercise he asks ...
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2answers
57 views

Logic problem: Atiyah-Macdonald 1.11

Proposition 1.11 in Atiyah-Macdonald's "Introduction to commutative algebra" states the following: "Given an ideal $I$ in a ring $A$ and $p_1, \dots p_n$ prime ideals, then $I \subset \cup_i p_i$ ...
0
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1answer
32 views

Why this application is well-defined?

Let $C$ be a curve. An application $\phi:C\to \mathbb P^n$ is called regular in a point $P\in C$ if there are regular functions $f_0,\ldots,f_n$ defined in a neighborhood $V$ of $P$ in $C$ such ...
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1answer
32 views

Compactness of Lie groups

Let $G$ be a Zariski-closed subgroup of $GL(V)$, where $V$ is an $n$-dimensional complex vector space. Question. Does $G$ have the structure of a compact Lie group? Such $G$ certainly is a Lie ...
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on special Kähler manifolds

Take Lie group $G$ with some hypotheses (compact, connected, maybe semisimple); call its Cartan subalgebra $\mathbf t \subset \mathbf g$ and Weyl group $W$. Why does the construction $\mathcal ...
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1answer
15 views

About freeness of modules over the coordinate ring of an affine variety

Let $X$ be an irreducible affine variety, $A$ be its coordinate ring, $M$ be an $A$-module. Suppose that for any maximal ideal $m$ of $A$, the localization $M_m$ is a free module of rank $n$ (finite ...
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1answer
19 views

A set $S\subseteq\mathbb{A}^n$ is quasi-affine iff $S=Z\setminus V$ for closed $Z$ and $U$?

I'm confused by a remark in note I'm reading. It essentially says, Let $S\subseteq\mathbb{A}^n$ be a subset of affine $n$-space over an algebraically closed field. It's clear that $S$ is ...
1
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1answer
43 views

Image of an arbitrary map falling on a algebraic set - criterion?

Let $f$ be a "typical" smooth non-polynomial map from $\mathbb{R}^3$ to $\mathbb{R}^7$. Is it reasonable to expect that $f(\mathbb{R}^3)$ is not included within the zero-set of a system of polynomial ...
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0answers
45 views

Is restriction of scalars an exact functor?

For the notion of restriction of scalars (aka Weil restriction) I have in mind, see this wiki page. My question then is Is the restriction of scalars an exact functor for ses of smooth linear ...
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0answers
36 views

Picard group schemes of degree d

Let $C$ be a smooth curve. I know that $Pic^0(C)$, i.e. the Picard group of degree 0 line bundles on $C$, is isomorphic to the jacobian $J(C)$, so it is an abelian variety. My question is, what about ...
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How could we define a sheaf or presheaf of polynomials? [closed]

Good evening everyone , Is there a sheaf or presheaf whose sections are polynomials defined on opens of a topology ? . If yes , what is this topology ?. Is it the Zariski topology , and why? And how ...
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1answer
36 views

Dimension and morphism with finite fibers

I'm studying the dimension of projective varieties and in the literature I'm reading I have the following statement: "If $f : X → Y$ is a morphism with finite fibers, i. e. such that $f^{−1}(P)$ ...
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67 views

Geometric Proof for Fermat's Last Theorem - A Question [closed]

I have been working on a geometric proof for Fermat's last theorem that I just realized has been worked on already in some shape or form (ba-dum-tsh). Before anyone says it, yes, I am aware that this ...
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0answers
36 views

algebraic varieties with log terminal singularities

I am looking for some non-trivial examples of algebraic varieties which have log-terminal singularities.
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0answers
25 views

Meaning of statement about differential forms

Let $S$ be a complex algebraic surface (smooth and proper over $\mathbf{C}$) and let $D$ be a divisor on $S$. What does it mean for a meromorphic section of the sheaf $(\Omega^1_S)^{\otimes m}$ to be ...
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0answers
27 views

Intersection numbers on product surfaces

Let $C_1$ and $C_2$ be smooth, projective curves over a field $K$. Let $S = C_1 \times C_2$. Let $D$ and $D'$ be (reduced) divisors on $S$ which map dominantly to both $C_1$ and $C_2$. How does one ...
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1answer
42 views

$(-1)$-curves and base change along a field automorphism.

Suppose that $E\subseteq S$ is a $(-1)$-curve inside a non-singular complex projective surface. By a $(-1)$-curve $E$, I mean that $E\cong\mathbb P^1_\mathbb C$ and $E^2=-1$. Now consider a field ...
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0answers
21 views

A question in K. Hulek, Elementary algebraic geometry

I'm reading Elementary Algebraic Geometry of Klaus Hulek, and I have a minor question about a proof of Proposition 1.62 in page 48-49. At the end of the proof, $W_i^v = \{w \in W:(v,w)\in Z_i\}$ is ...
2
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2answers
76 views

Hartshorne II Prop 6.8

My weaknesses with commutative algebra are really slowing down my progress through Hartshorne. I hope someone can help me understand some statements in the proof of the proposition below. Prop ...