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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2
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1answer
25 views

Maximal tori in $SO(n,\mathbb{C})$

What are maximal tori in $SO(n,\mathbb{C})$? (not $SO(n,\mathbb{R})$) Can a maximal torus in $SO(n,\mathbb{C})$ be written as $T\cap SO(n,\mathbb{C})$ for some maximal torus $T$ in ...
0
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1answer
42 views

Manifolds as homology classes

I have found that a k-dimensional submanifold of a manifold M can be considered as a class in the homology group $H_{k}(M)$. Why ?
0
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2answers
52 views

Is there a general way to parameterize all implicit functions?

We all know some curves can be described by $y=f(x)$ and some surfaces can be described by $z=f(x,y)$ However, there exists curves and surfaces which cannot be described by those, such as a circle and ...
6
votes
1answer
206 views

Prove that $k[x,y,z,w]/(xy-zw)$, the coordinate ring of $V(xy-zw) \subset \mathbb{A}^4$, is not a unique factorization domain

I want to show that $k[x,y,z,w]/(xy-zw)$, the coordinate ring of $V(xy-zw)\subset\mathbb{A}^4$, is not a unique factorization domain. Morally, all we need to do is find some nonzero element that ...
0
votes
1answer
46 views

Completion of quotient of polynomial ring

Hartshorne's Algebraic Geometry uses the following facts on page 35 without proof: The completion of $(k[x,y]/(y^2-x^2-x^3))_{(x,y)}$ is $k[[x,y]]/(y^2-x^2-x^3)$ and that of ...
0
votes
1answer
54 views

Castelnuovo-Mumford regularity and the maximal degree of generators

I am reading a few texts on Castelnuovo-Mumford regularity. If I understand correctly, almost all of them say: If $I$ is a homogeneous ideal in $k[X_0,...,X_n]$ where $k$ is algebraically closed ...
0
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0answers
62 views

On the phrase “identify the graph…”

Let $\psi\colon \mathbb{P^1} \to \mathbb{P^1}$ be an isomorphism; identify the graph of $\psi$ as a subvariety of $\mathbb{P}^{1} \times \mathbb{P}^{1} \cong Q \subset \mathbb{P}^{3} $. Now do the ...
0
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0answers
39 views

Pushforward of a volume form

Let $X$ be a complex projective manifold with semi-ample line bundle $ K_X$ . Assume that $f: X\to X_{can}\subset \mathbb CP^N$ , and $f^{-1}(s)$ is nonsingular fibre, then I am looking for a proof ...
2
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0answers
37 views

Regular function on a variety which is not globally rational

I am looking for a particularly simple example of a regular function $f : V \to \mathbb{A}^1_k$ for some affine variety $V \subseteq \mathbb{A}^n_k$ over a field $k$, which cannot be expressed by a ...
0
votes
1answer
18 views

Birational map between manifolds

I have to show that the manifold $A=\{ [z_{1}:z_{2}:z_{3}:z_{4}]\in \mathbb{C}\mathbb{P}^{3} | z_{1}z_{3}^{n} - z_{2}z_{4}^{n}=0\}$ is birational equivalent to $\mathbb{C}\mathbb{P}^{2}$, how can I ...
10
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1answer
220 views

Sard's theorem for algebraic varieties

(One version of) Sard's theorem states that: Theorem (Sard): Given $M$ and $N$ smooth manifolds of dimensions $m$ and $n$ respectively, and a smooth map $f:M\to N$, then the set of singular values ...
3
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0answers
41 views

sections of birational proper morphism over an etale cover

Let $f: Y \to X$ be a birational proper morphism. Assume that every point of $X$ has an etale neighbourhood over which $f$ has a section. Is it true that $f$ is an isomorphism?
3
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0answers
99 views

Ring of rational power series

Let $A$ be any commutative ring with 1. A power series $f\in A[[t]]$ is called rational if we can find a $g\in A[t]$ such that $fg\in A[t]$. It is clear that the set of rational power series forms a ...
0
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2answers
44 views

question on quadric hypersurfaces

Over $\mathbb{C}$, every homogeneous polynomial of degree $2$ in $x_0,...,x_n$ can be brought into the form $f=x_0^2+...+x_r^2$ for some $0\le r\le n$. This is a part of an exercise of Hartshorne's ...
1
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0answers
38 views

Weil restriction for schemes

I'm trying to understand the Weil restriction of a scheme (since I'm reading a paper which uses it). I'm even having troubles trying understanding the following "toy" example. Toy example. Let $X$ be ...
1
vote
0answers
15 views

Is there any bound on the number of generators of a monomial ideal in C(x,y)? [closed]

Just the question in the title, also if there is such a bound, say what it is.
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0answers
34 views

Is it important to study plane algebraic curves before read Fulton's book

I'm studying Fulton's algebraic curves book and I would like to know how important study plane algebraic geometry before read Fulton's book. Example of books on this subject: Algebraic Curves - ...
0
votes
1answer
33 views

Sign of first chern class with some conditions

Let $X$ be a compact Kahler algebraic variety which $K_M$ is big and nef, and $Kod(X)=dimX$ then why the first chern class $c_1(M)$ is negative or zero . I don't undrestand kawamata's theorem in this ...
0
votes
1answer
31 views

What does linearly equivalent mean in this context

I'm trying to understand this proof of Fulton's algebraic curves book page 107: I didn't understand what does linearly equivalent mean in this context and why this implies it suffices to show that ...
3
votes
0answers
46 views

Definition of the $\Bbb C^*$-weight of a line bundle

I'm new to geometric invariant theory and am unsure about a definition. Let $X$ be a smooth projective-over-affine variety equipped with a $\Bbb C^*$ action and let $Z$ be the fixed locus of this ...
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0answers
15 views

Characterization of ideals generated by homogeneous polynomials in terms of $f^{(d)}$ in Gathmann's notes.

On pg. 37 of Gathmann's Algebraic Geometry notes, the following is mentioned: For every $f\in k[x_0,x_1,\dots,x_n]$ be an ideal. The following are equivalent: I can be generated by ...
0
votes
1answer
48 views

Why this order is well-defined?

I'm trying to understand why this order defined in Fulton's algebraic curves book is indeed well-defined: I have two questions: Why $u\in \mathcal{O_P}(X)$ (he is implicitly assuming this fact so ...
4
votes
0answers
62 views

Openness of a subset in complex 2-plane

Let $U$ be a subset of $\mathbb{C}^{2}$ containing the origin $0$. Assume that for any curve $C$ (an affine variety of dimension 1, maybe singular) passing through $0$ we have $U \cap C$ is ...
2
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0answers
52 views

Are there any fundamental new discoveries in Algebraic Number Theory than in ‘traditional’ number theory ? [closed]

I am new to Algebraic Number Theory. I wonder if there is any fundamental new discoveries in Algebraic Number Theory than in ‘traditional’ number theory ? I want to know, beside ‘generalizing’ or ...
1
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0answers
32 views

Does the canonical morphism commute with direct image functor?

I am trying to prove the representability of the Quotient functor. I have the following problem. Let $\phi \colon T \to S$ be a morphism of noetherian schemes and let $F$ be a coherent sheaf on ...
0
votes
0answers
33 views

Scheme of Sections of a Coherent Sheaf

Suppose given a flat, projective morphism of finite type noetherian $\mathbb{C}$-schemes $X \rightarrow T$ and a coherent sheaf $M$ on $X$. Define a contravariant functor $F:Sch/T \rightarrow Grp$ ...
3
votes
0answers
38 views

Resultants of two polynomials over a ring

Let $k$ be a field $f,g\in k[x,y]$ be two polynomials. The resultant $R\in k[x]$ is a polynomial function of the coefficients of $f$ and $g$, such that $f$ and $g$ gave a common zero (in an extension) ...
0
votes
1answer
27 views

Rational first chern class of algebraic variety with zero Kodaira dimension.

Let $X$ be a compact Kahler algebraic variety which has zero Kodaira dimension. Then the integral first chern class vanishes? What about rational first chern class?
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0answers
27 views

A question about multiplicites of points on a curve.

Shafarevich says the following: If $P=(0,0)$ and the leading terms of the equation of the curve have degree $r$, then $r$ is called the multiplicity of $P$, and we say that $P$ is an r-tuple ...
6
votes
1answer
189 views

Are varieties of Kodaira dimension zero precisely the varieties with torsion canonical sheaf

Let $B$ a smooth projective connected variety over $\mathbf C$. Suppose that $K_B$ is torsion. Then, clearly, the Kodaira dimension of $B$ is zero. Does the converse hold? That is, suppose that $B$ ...
1
vote
1answer
32 views

A question about Lüroth's theorem.

Shafarevich says the following: Using Lüroth's theorem, we see that if $X$ is a rational curve, then $k(X)$ is isomorphic to the field of rational functions $k(t)$. This is equivalent to saying ...
2
votes
0answers
32 views

A corollary in Kollar's paper

I'm reading Kollar's paper: Toward moduli of singular varieties Compositio Mathematica, tome 56, no3 (1985), p. 369-398. ...
2
votes
1answer
35 views

Sheaf of sections vanishing at a point is $\Gamma(E) \otimes I_p$

Notation: $E$ is a vector bundle with sheaf of sections $\Gamma(E)$. $I_p$ is the sheaf of regular functions on the base space vanishing at a point $p$. $\Gamma_p(E)$ is the sheaf of sections of $E$ ...
2
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0answers
52 views

What do I need to understand this article

I've just finished Fulton's algebraic curves book and I would like to know what do I need to know to understand this article: Weierstrass semigroups and the canonical ideal of non-trigonal curves. I ...
4
votes
1answer
82 views

Problem I.5.4(c) in Hartshorne

The problem asks to show that if $Y$ is a projective curve in $\mathbb{P}^2$ of degree $d$ and $L$ is a line such that $Y \neq L$, then $\sum_{P \in L \cap Y} (L \cdot Y)_P = d$. The solution given ...
2
votes
1answer
247 views

How can we prove that formal smoothness is a property local on the source?

I have learned from this question that, in spite of the gap in the proof of 17.1.6 (i) in EGA IV, we can still verify that a morphism of schemes is formally smooth locally on the source. But, even ...
1
vote
3answers
113 views

Some practical questions on cohomology and the ring $\mathbf{Z}[x]/(x^2)$

So I know that the cohomology ring of $S^n$ is $\mathbf{Z}[x]/(x^2)$ with "$x$ in degree $n$"; if we ignore the grading then this ring fails to distinguish the spheres. What is actually meant by "in ...
2
votes
4answers
197 views

$\mathbb{C}[x,y]/(f,g)$ is an artinian ring, if $\gcd(f,g)=1$. [closed]

This problem extends the fact that $\mathbb{C}[x,y]/(x^n,y^m)$ is artinian ring. Let $f,g \in \mathbb{C}[x,y]$ such that $\gcd(f,g)=1$. Show that $\mathbb{C}[x,y]/(f,g)$ is an artinian ring.
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1answer
38 views

When is $k(X)$ algebraic over $k(Y)$ for a dominant morphism $f:X\rightarrow Y$ between varieties.

Let $f:X\rightarrow Y$ be a dominant morphism between irreducible varieties over an algebraically closed field $k$. When is $k(X)$ algebraic over $k(Y)$? Is there an if and only if criterion? What if ...
2
votes
1answer
261 views

Lines in $\mathbb{A}^3$

This seems intuitive, but I'm having trouble coming up with an exact matrix for the problem. Let $\{L_1, \ldots, L_N\}$ be a set of lines through the origin $(0,0,0)$ in the affine space ...
5
votes
3answers
57 views

Morphism of schemes $f\colon X\to Y$ associated to a continuous map of the underlying spaces $|X|\to |Y|$

I am sorry for asking two questions in one but they are strongly related. What is an example of (affine?) schemes $X=(|X|,\mathcal{O}_X)$ and $Y=(|Y|,\mathcal{O}_Y)$ and a map of topological spaces ...
2
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4answers
57 views

Find the center of a circle on the x-axis with only two points, no radius/angle given

Find the center $C$ on the x-axis of the circle containing $(15,-2)$ and $(7,10)$ I can't seem to find a formula to help me solve this problem without needing the radius or the angle between the the ...
0
votes
1answer
36 views

transversal surfaces

I have to prove that the surfaces $E=\{([0:x_{1}:x_{2}],[y_{1}:y_{2}])\in\mathbb{P}\mathbb{C}^{2}\times\mathbb{P}\mathbb{C}\}$ and $V=\{([x_{1}:0:x_{2}],[0:y_{2}])\in \mathbb{P}\mathbb{C}^{2} \times ...
2
votes
1answer
98 views

Help in this notation in Fulton's Algebraic Curves book

I'm reading Fulton's Algebraic Curves book, I'm stuck in the following proposition (page 105): In fact, what I didn't understand is the following notation in the proof of this proposition: Why ...
4
votes
0answers
49 views

On the definition of groups of multiplicative type

Let $k$ be a field of characteristic 0. The definition of a linear algebraic $k$-group of multiplicative type (m.t.) I've seen the most in the literature is that $G$ is of m.t. if it is a ...
7
votes
2answers
188 views

Showing that $x^3+y^3+z^3=0$ is not rational

Is there a short proof that $F:x^3+y^3+z^3=0$ in $\mathbf{P}^2$ is not rational, apart from using the genus? Perhaps this is an elliptic curve, so every morphism $\mathbf{P}^n\rightarrow F$ is ...
1
vote
1answer
53 views

Projective curve $x^d+y^d+z^d=0$ is nonsingular using Jacobian matrix

According to this question: Nonsingular projective variety of degree $d$, the curve $x^d+y^d+z^d=0$ in $\mathbb{P}^2$ is nonsingular. I'm trying to prove this. Hartshorne defines nonsingular ...
0
votes
1answer
53 views

Proof that presheaf is a sheaf for Spec

Atiyah Macdonald define presheaf (chapter 3, exercise 23) on the base of $Spec(A)$, where $A$ is commutative ring with $1$, as follows $$ \mathfrak{F}(X_f) = A_f, $$ where $X_f$ is a basic open set ...
3
votes
0answers
35 views

Zariski cohomology of $\mathbb{A}^1$ over a local ring with values in $\mathbb{G}_m$

Let $X$ be a the spectrum of a regular local ring. What is known about the vanishing of the Zariski cohomology group $$ H^n(\mathbb{A}^k_X,\mathbb{G}_m) $$ for $n,k\geq 0$? If $X$ has dimension $d$ ...
0
votes
1answer
59 views

Quotient of locally free sheaf is locally free?

If $0\rightarrow F\rightarrow G\rightarrow H \rightarrow 0$ is an extension of $\mathcal{O}$-modules with $F$ and $G$ locally free (each of constant finite rank, i.e. vector bundles), then is $H$ ...