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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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
21 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
26 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
21 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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8 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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27 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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12 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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17 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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55 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
32 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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41 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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23 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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35 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
44 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
28 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
68 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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43 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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21 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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24 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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29 views

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
48 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 algebraic geometry Ex 7.4E. Ex7.4E: Assume Chevalley's thm.Show that a morphism of affine k-varieties $\pi:X \rightarrow Y$ is surjective iff it is surjective on ...
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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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30 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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36 views

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
35 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
56 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 $I \subset \cup_i p_i$ ...
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1answer
29 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
30 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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35 views

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
8 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
17 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 ...
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1answer
42 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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43 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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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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40 views

How could we define a sheaf or presheaf of polynomials? [on hold]

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
35 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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65 views

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

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
27 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
24 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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26 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
38 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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18 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 ...
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2answers
69 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 ...
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43 views

A question about an exercise in Klaus Hulek book

"Let $X, Y \subset \mathbb{C}^4$ be varieties defined by $$ X := \{ (t,t^2,t^3,0) \,|\, t \in \mathbb{C} \}, \quad Y := \{ (0, u, 0, 1) \,|\, u \in \mathbb{C} \}. $$ The join variety of $X$ and $Y$ ...
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2answers
20 views

If $\mathcal{I}(-)$ is the ideal map on subsets of affine space, why does $A\subseteq\overline{B}\iff\mathcal{I}(B)\subseteq\mathcal{I}(A)$?

I think this is a basic property of $\mathcal{I}(-)$, but I'm having trouble seeing it. I denote by $\mathbb{A}^n$ the affine $n$-space over an algebraically closed field $k$, where if ...
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2answers
26 views

Canonical embedding

I'm reading the second chapter of this paper and I need help to understand what exactly this canonical embedding is: Remark: $C$ is a smooth non-hyperelliptic complete irreducible algebraic curve ...
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1answer
22 views

A question about Klaus Hulek algebraic geometry (regarding Noether normalization)

This is the proof of Noether normalization on p.30 of Klaus Hulek's elementary algebraic geometry. And on the next page, the book says that "Analyzing the above proof, we see that y1, .., ym can be ...