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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19 views

Is there genera for algebraic cobordism?

For real and complex manifolds, we can form the (oriented) cobordism ring $\Omega$, and a genus is defined to be a ring homomorphism $$\varphi:\Omega\otimes\mathbb{Q}\to R$$ where $R$ is any ...
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9 views

Groebner basis over rings

Let $I$ be an ideal in $A[x_1, \ldots, x_n]$, where $A$ is a Noetherian commutative ring, such that w.r.t some monomial order it has a Groebner basis $G = \{g_1, \ldots, g_t\}$ with all the leading ...
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12 views

Covering number of the set of $n_1\times n_2$ matrices of rank at most $r$

What is the covering number of the set of $n_1\times n_2$ matrices of rank at most $r$? We know that the dimension of the set is $r(n_1+n_2-r)$. Thus, the covering number $N(\rho)\le C ...
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0answers
19 views

Adelic definition of “canonical divisor”

For a function field over a curve $F/K$, some book define the canonical divisor as the divisor of a map $\omega:\mathscr{A}_{F}\rightarrow K$ (where $\mathscr{A}_{F}$ is the pre-adele, ie. adele but ...
2
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0answers
41 views

Homological description of the degree of a map to $\mathbb P^n$

Let $f \colon X \to \mathbb P^n$, $n \geq 2$, be a holomorphic map from a compact Riemann surface $X$ and whose image $f(X)$ is a smooth projective curve. There are two notions of degree for such a ...
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29 views

Base change and relative dimension

Let $B$ a ring of Krull dimension $1$ and suppose that $\text{Spec }B$ is irreducible. And let $f:X\to\text{Spec}(B)$ a $B$ scheme with the following properties: $X$ is projective, regular and ...
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27 views

Does this theorem hold for real varieties? + Reference request - Real varieties.

Let $X$ be a complex variety and let $X_\mathbb R$ be its real part, that is the $X_\mathbb R$ consists of all the real valued points of $X$. I would like to learn a little more about the real ...
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0answers
27 views

identity for quaternions' group

Could you help me for solving this: Let $Sp(n)$ be the group of linear transformations of $H^n$ such that preserve hermitian form $$\sum_{i=1}^n \overline{q_i}r_i,$$ that $H$ is the quaternions _ the ...
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0answers
27 views

Rational sections of invertible sheaves and hermitian inner products

Notations: Let $X$ be a $\mathbb C$-scheme of finite type, projective, integral and of dimension $1$ (i.e. an algebraic curve) and with function field $K(X)$. The set of closed points is $X(\mathbb ...
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0answers
23 views

Derivative of the Klein j-invariant

To prove that the field of meromorphic functions on $X(1)$ is generated by the Klein j-invariant, we need to show that the derivative of $j(\tau)$ does not vanish. (It only has simple zeroes). But I ...
2
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1answer
29 views

Can a separable isogeny of elliptic curves have an inseparable dual?

Let $\phi: E_1\to E_2$ be an isogeny of elliptic curves over a field $K$ of characteristic $p>0$. Suppose that $\phi$ is separable and let $\hat{\phi}: E_2\to E_1$ denote the dual isogeny. Then ...
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0answers
72 views

Intersection of algebraic curves at a point with given multiplicity

I don't know if this question is too basic for MO, so I put it here, but if you think I should migrate the question to MathOverflow please suggest me. Let $C/k$ be a smooth curve over a perfect ...
2
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0answers
35 views

Number of $\mathbb{F}_q$-rational points on a smooth variety

From the proof of Weil's conjectures it follows that $|q^k - \# X(\mathbb{F}_{q^k})| = O(q^{k(n - \frac{1}{2})})$, where $X$ is a smooth variety over $\mathbb{F}_q$ and $n = \dim X$ (see for example ...
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0answers
30 views

reference for “wonderful compactification”

I am trying to learn about the wonderful compactification for (adjoint) semisimple groups. Are there any good references that sketches out the full construction other than ...
1
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1answer
23 views

why does the regular action of the structure group not imply triviality of a fibre system?

Let $P$ and $X$ be algebraic varieties, $\pi:P\to X$ a morphism and $G$ an algebraic group acting on $P$. Serre calls the triple $(G,P,X)$ a fibre system an proves that if it is locally isotrivial ...
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0answers
36 views

Kahler differentials over non-algebraically closed fields

Let $A = k[x_1,\ldots,x_n]/(f_1,\ldots,f_m)$ be a finitely-generated $k$-algebra, then at least when $k$ is algebraically closed, the module of Kahler differentials is $$ \Omega_{A/k} = A dx_1 \oplus ...
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29 views

Showing a curve is a variety (example)

I am working on Exercise 2.8 of Fulton's book on Algebraic Curves. Briefly, it asks to show $V = V(XZ-Y^2, YZ-X^3,Z^2-X^2Y)$ is a variety over $\mathbb{C}$. I tried using the hint there. We define ...
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0answers
24 views

Question regarding Geometric meaning of Noether normalization theorem for projective varieties

In Ernst Kunz's ''commutative algebra and algebraic geometry'' book, ch.2, proposition 4.5 the author states: The Noether Normalization Theorem admits the following application in projective ...
3
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1answer
44 views

Correspondence between vector bundles and locally free sheaves

I am trying to look at smooth manifolds in the context of locally ringed spaces. Vector bundles have a characterization in terms of sheaves of modules: if $X, \mathcal{O}_X$ is a topological (or ...
2
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1answer
64 views

Possible Inaccuracy at classic paper by Bayer and Stillman

In reading the paper Bayer and Stillman, "A criterion for detecting $m$-regularity", i believe i have encountered what may be a little inaccuracy, which i describe next. Let $I$ be a homogeneous ...
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1answer
30 views

Question on irreducibility of curves over $\mathbb{R}$

Suppose we look at the elliptic curve $y^2 = x^3 - x$ over $\mathbb{R}$. Then this has two connected components in the Euclidean topology because the cubic has three real roots. However, the ...
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2answers
40 views

Meaning of localization map in the structure sheaf of an affine scheme

I'm reading the book $\textit{The Geometry of Schemes}$ and am a bit confused about the definition of the structure sheaf of an affine scheme. For a ring $R$, we define the $\textit{distinguished ...
2
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1answer
55 views

a little “paradox” in local cohomology of zero-dimensional ideals

Let $S = k[x_1,x_2,x_3]$ be a polynomial ring of dimension $3$ over an infinite field, and let $I$ be a homogeneous ideal of height $3$. Since $S$ has no zero divisors, the Krull dimension of $I$ is ...
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1answer
64 views
+50

Proof of algebraic set involving dimension

I need some help to understand the following proof. Let $k$ a field and $V$ an algebraic set. I note $\mathfrak{m}_P$ the ideal generated by $X_1-a_1,\dots ,X_n-a_n$ in $k[V]=k[X_1,\dots ,X_n]/I(V)$ ...
2
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0answers
27 views

Proving subgroup of $Aut(\Bbb C^2)$ that fixes a specific curve is isomorphic to $\Bbb Z^6 \times \Bbb Z^3 $

So, I have the curve $C = V(y^3 - x^6 + y^6) \subset \Bbb C^2$. I want to prove that, if $G= \{ \varphi = (f_1,f_2) \in Aut(\Bbb C^2):\varphi(C) = C,$ $ deg(f_i) = 1 \}$, then $G \simeq \Bbb Z^6 ...
2
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1answer
46 views

Exercise II-11 from Eisenbud-Harris, subscheme of dimension $0$, degree $3$, supported at origin isomorphic to what?

Suppose that $K$ is algebraically closed, and let $Z = \text{Spec}\,K[x_1, \ldots, x_n]/I \subset \mathbb{A}_K^n$ be any subscheme of dimension $0$ and degree $3$, supported at the origin. How do I ...
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1answer
32 views

Every divisor $D$ on $X$ is dominated by a divisor linearly equivalent to $mA$

I am reading the proof of Riemann-Roch theorem from Shafarevich's Basic Algebraic Geometry 1 (3rd edition), but I'm stuck on a Lemma on pg 215. It says (II) Every divisor $D$ on $X$ is dominated ...
3
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1answer
63 views

Do finite morphisms preserve dimensions?

If $f: Y\to Z$ is a finite, surjective morphism of normal integral schemes (of finite type over a field) and $y$ is a prime divisor of $Y$, is then also $z= f(y)$ of codimension 1? We have an ...
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2answers
36 views

Quadratic equations defining the $10$-dimensional spinor variety.

Let $S$ be the $10$-dimensional Spinor variety parametrizing one of the two families of $4$-dimensional linear subspaces of the non-singular quadric in $\mathbb{P}^{9}$. I have read that there exist ...
2
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1answer
62 views

Show that Riemann Surface is connected?

I was reading Artin's Alegbra when this question came into my mind. Consider $f(t,x)=x^{2}-t$ , The locus X of zeros in $\mathbb C^{2}$ of a polynomial is called Riemann surface of f. I understood ...
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0answers
49 views

Quick question: Line bundle on union of two lines

Let $l_1$, $l_2$ be two lines in $\mathbb{P}^n$. What is the meaning of $\mathcal{O}_{l_1}(a_1)\cup\mathcal{O}_{l_2}(a_2)$ as a sheaf on the union $C=l_1+l_2$ of two distinct lines and why do we ...
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0answers
28 views

If $\pi : X \to Y$ is a flat, proper $O$-connected morphism of locally Noetherian schemes, then is $h^0(X_q, O_{X_q}) = 1$?

If $\pi : X \to Y$ is a flat, proper $O$-connected morphism of locally Noetherian schemes, then is $h^0(X_q, O_{X_q}) = 1$ (the dimension of the space of global sections of the fibers)? This seems to ...
0
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1answer
59 views

Short exact sequences from the Euler sequence.

I was reading an article in which the author said that the sequence $\require{AMScd}$ \begin{CD} 0 @>>>\Omega ^1_{\mathbb{P}^n} @>>> \mathcal{O}_{\mathbb{P}^n}(-1)^{\oplus n} ...
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1answer
45 views

Finding the maximal ideals of the quotient of a polynomial ring by an ideal

Let the field $k$ be algebraically closed, let $f(X) \in k[X]$ be a separable polynomial of degree at least $2$, let $$ B = \frac{k[Y,X]}{(Y^2 - f(X))} $$ and write $y,x$ for the images in $B$ of $Y$ ...
3
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0answers
53 views

About birational map between curves of the form $y^{2}=g(x)$

I'm trying to solve the following exercise but I'm stuck after trying for a long time. Suppose that $g(x)=ax^{4}+bx^{3}+cx^{2}+dx+e\in{k[x]}$ and similarly ...
2
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0answers
33 views

Extension of Leray spectral sequence, Vakil's 23.4 H

If you have a morphism \begin{equation} (X,\mathscr{O}_X) \xrightarrow {\pi} (Y, \mathscr{O}_Y) \end{equation} for every $\mathscr{O}_X$-module $\mathscr{F}$, there is a spectral sequence with $E_2$ ...
2
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1answer
29 views

tangent space of a curve in projective space

Suppose we have the curve $Z\subset \mathbb{P}^2$ given by the equation $y^2z-x^3=0.$ I have to find a basis for the tangent space at $(0:0:1)$, but I find the definition hard: Let $X$ be a variety ...
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0answers
9 views

How to determine the angle from a point and the plane tangent points in a sphere

I have an UAV modeled in three dimensions with let's say position coordinates $p_{uav} = (x_1,y_1,z_1)$ that is moving in a direction $d = (d_x,d_y,d_z)$ and a moving obstacle modeled as a sphere with ...
3
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0answers
68 views

Proof of Chapter 2 Proposition 2.6a in Silverman Arithmetic of Elliptic Curves

The following is Proposition 2.6(a) in Silverman's AEC: "Let $\phi: C_1 \rightarrow C_2$ be a nonconstant rational map of smooth (projective) curves over an algebraically closed field $K$. Then ...
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2answers
63 views

Fibers of extension of scalars to algebraic closure of an affine variety is of dimension 0.

Let $X$ be an affine variety over $k$ and $\overline{k}$ be the algebraic closure of $k$. Then the projection morphism $X_\overline{k} = X \times _k \overline{k} \to X$ has dimension 0 fibers. ...
3
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1answer
54 views

If a divisor $D$ satisfies that $D^{2}=1$, is it true that the morphism induced by $|D|$ is birational?

Let $X\subset \mathbb{P}^{5}$ be a non-degenerate algebraic surface. Let us suppose that $D\subset X$ is a curve such that $D^{2}=1$. I would like to know if the rational map induced by the complete ...
1
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1answer
15 views

Number of zero-solutions for two bivariate polynomials $p$ and $q$

If I consider two bivariate polynomials $p,q \in \mathbb{C}\left[ x,y \right]$ where $p$ has total degree $m$ and $q$ has total degree $n$. To keep things simple I'm not interested in special cases ...
0
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0answers
22 views

Self-intersection of an axis

Let $X$ be a projective, smooth curve over an algebraically closed field, $Y = X\times X$ and $\mathcal{l}= pt\times X$ where $pt$ is a closed point of $X$. Can one say that the intersection number ...
1
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1answer
33 views

Ideal ring of polynomials in two variables with real coefficients

Let $ I= \langle x ^ 4 + y ^ 4 + 2x ^ 2y ^ 2-x ^ 2-y ^ 2 \rangle \subset\mathbb R[X,Y]$. I want to determine whether $ I $ is prime or radical. I know that $I$ is not prime. First, $ \langle x ^ 4 + ...
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0answers
23 views

Brill-Noether theory- reference request

I need some reference, some books or something suitable for a begginer. I found some .pdf's on google, that have interesting introduction, but couldn't find any book.
0
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1answer
37 views

Relation between Homeomorphisms and Isomorphisms for varities.

I am right now learning Algebraic Geometry and at the first moment is very demanding. One of my biggest doubts is: why algebraic geometers despise so much homeomorphisms , all books that I have been ...
0
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0answers
29 views

Definition of hypersurface singularity

I am really confused about this notion. Suppose $X$ is an arbitrary variety over an algebraically closed field $k$ (if you like, let the characteristic be $0$), and $p$ is a $k$-valued point. If $p$ ...
0
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1answer
36 views

Page 276 of Principles of Algebraic Geometry by Griffiths and Harris; wrong parameter count?

The following is taken from page $276$ of Principles of Algebraic Geometry by Griffiths and Harris: Now let $S$ be a Riemann surface of genus $g\ge 3$. By our last result, if $S$ has any ...
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0answers
22 views

Volume question regarding segmenting a truncated cylinder.

Picture of a truncated wedge segment 2If you segmented a truncated cylinder, ensuring all segments had the same volume, where would the intersection be? I'm understand there'll probably not be a ...
-2
votes
1answer
50 views

What exactly is the $O_X$-module and the corresponding sheaf of modules?

I am very puzzled by the definition in the Wiki page. I understand that over a subset $U$ we can assign a sheaf of abelian groups, e.g. some analytic functions over $U$. So we consider that these ...