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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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?
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25 views

Does “toric” conflict with “Calabi-Yau” in the projective case?

Let $X$ be a Calabi-Yau complex algebraic variety. If it is projective, we can talk about its geometric genus $p_g=h^{\dim X, 0}$, and the Calabi-Yau condition says that $p_g=1$. Now, one might be ...
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2answers
36 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 ...
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0answers
13 views

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

Just the question in the title, also if there is such a bound, say what it is.
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0answers
30 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 - ...
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1answer
30 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 ...
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0answers
12 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 ...
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0answers
44 views

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

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 ...
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1answer
46 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 ...
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0answers
26 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
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1answer
26 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 ...
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0answers
27 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 $F:Sch/T \rightarrow Grp$ ...
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0answers
34 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) ...
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0answers
26 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 ...
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0answers
32 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 ...
0
votes
1answer
24 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
31 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. ...
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1answer
30 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 ...
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0answers
53 views

An estimate involving gaps in a subsemigroup of $(\mathbb N,+)$

I think this question can be solved by a high school student, maybe there is some trick on it or I'm forgetting something. Before my question, some background is required: Definition: A ...
2
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1answer
31 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 $E$ ...
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0answers
55 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
51 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 ...
2
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2answers
71 views

When is a polynomial map proper?

Let $f\in \mathbb C[x_1,\dots,x_d]$ be a (nonconstant) polynomial. Of course it can be viewed as a (surjective) regular map $$\tilde f:\mathbb A^d_\mathbb C\to \mathbb A^1_\mathbb C.$$ Question. ...
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1answer
36 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 ...
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3answers
54 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 ...
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votes
0answers
92 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 ...
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1answer
34 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
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1answer
96 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 ...
0
votes
1answer
52 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 ...
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1answer
49 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 ...
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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 ...
3
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0answers
34 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 $d$ ...
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0answers
38 views

Direct image of the exceptional divisor along a blow-up

Let $X=\mathrm{Spec}(k[x_1,\ldots,x_n])$ for $n\geq 2$, and let $\mathcal{I}=\widetilde{I}\subseteq\mathcal{O}_X$ for an ideal $I\subseteq k[x_1,\ldots,x_n]$. Let ...
4
votes
1answer
42 views

If $G/H$ and $G$ are connected linear algebraic groups must $H$ also be connected?

Let $k$ be a perfect field (e.g of characteristic zero) and let $G$ and $H$ be linear algebraic groups over $k$, with $H$ a normal subgroup of $G$. If both $G$ and $G/H$ are connected, must $H$ ...
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votes
1answer
23 views

A Small Problem on the Surjectivity of a Map Between Two Locally Free Sheaves

I met a problem when reading Positivity In Algebraic Geometry I. It is Example $1.8.15$ in Chpater 1 Section 8, which is called Green's Theorem. Let $W$ be a subspace of $H^{0}(P, O_{P}(d))$, and $W$ ...
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0answers
36 views

differential of $f:X\to\Sigma$ as an elliptic surface,

Let $X$ be an algebraic surface surface and $\sum$ an algebraic curve, and assume, $f:X\to\Sigma$ be an elliptic surface, my question is Why the differential $df$ can be viewed as an injection of ...
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1answer
37 views

finding $\lambda$ when equation of parabola is given

If the equation $\lambda x^2 + 4xy + y^2 + \lambda x + 3y + 2 = 0$ represents a parabola. Then find $\lambda$. I got stuck in this question while solving parabola. Is here anybody who can help me ...
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0answers
45 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 ...
3
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1answer
87 views

Geometrical description of maps of schemes

In preparation for an exam, I am trying to solve the following question: Describe geometrically all maps from $\operatorname{Spec}(\mathbb{C}[z]/(z^2))$ to $\operatorname{Spec}(\mathbb{C}[x,y])$. ...
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0answers
21 views

Graph of a regular function. When is the projection on the first component birational?

Let $X$ be an irreducible variety over a field $k$ and $f$ a regular function on some open subset $U\subseteq X$. Let $F\subseteq X\times \mathbb{A}^1_k$ be the graph of $f$ and suppose that the graph ...
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0answers
28 views

Definition - Logarithmic Zeros and Algebraic Vector Fields

In Algebraic Geometry, what is the difference between an algebraic vector field and a derivation (are they completely different or synonymous)? What does it mean to say an algebraic vector field has ...
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2answers
46 views

Map from $\mathbb{C}^2$ to $\mathbb{A}_{\mathbb{Q}}^2$

I was doing Exercise 3.2 I on Ravi Vakil's notes on Algebraic Geometry: consider the map of sets $\phi: \mathbb{C}^2 \rightarrow \mathbb{A}_{\mathbb{Q}}^2$ defined as follows. $(z_1, z_2)$ is sent to ...
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votes
1answer
53 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 $H$ ...
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166 views

Looking for a smooth curve that is not rational

I am preparing for an exam in (mostly classical) algebraic geometry, and I have some preparatory questions, among which: Can you write the equations of any nonsingular curve in any projective ...
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1answer
39 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 $(k[x,y]/(xy))_{(x,y)}$ is ...
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0answers
42 views

What does “$\overline{G}_*$ is the residue of $G_*$ in $\mathscr{O}_P(F)$” mean in Fulton's book on algebraic curves?

I'm trying to understand this phrase in Fulton's algebraic curves book page 53: Anyone could help me? Thanks
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1answer
62 views

Can $xy=0$ be the image of an algebraic morphism $\mathbb A^2 \rightarrow \mathbb A^2$?

Suppose we have an algebraic morphism $f:\mathbb{A}^2\rightarrow \mathbb{A}^2$. Can the image of $f$ be the zero locus of the polynomial $xy$? I think not, at least not if we're working over ...
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1answer
28 views

Help in this characterization of the gaps of the symmetric numerical semigroups

Before my question, some background: Definition 1: A numerical semigroup is a subsemigroup $N$ of the additive semigroup $\mathbb N$ of the non-negative integers such that $\mathbb N-N$ is ...
7
votes
2answers
178 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 ...
4
votes
1answer
141 views

$k$-algebra homomorphism of the polynomial ring $k[x_1,\dots,x_n]$

Let $\phi:k[x_1,\dots,x_n]\mapsto k[x_1,\dots,x_n]$ be a $k$-algebra homomorphism with $\phi(x_i)=f_i$, where $k$ is algebraically closed and has characteristic zero. I have the following questions: ...