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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16 views

Intuition about formal brances of a curve at a point

Consider an algebraic surface $X$ and a curve $Y\subset X$. Here $X$ is a $K$-scheme integral of finite type of dimension $2$ and $Y$ is a closed subscheme of dimension $1$. Fix a closed point ...
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2answers
45 views

Is the ideal of a variety the annihilator of a subspace of the symmetric algebra?

Let $V$ be a vector space over an algebraically closed field $K$. Let $\mathrm{Sym}(V^*)=\mathrm{Sym}(V)^*$ be the symmetric algebra on $V$, i.e. if we give a basis $e_1,...,e_n$ of $V$ and let ...
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0answers
29 views

Linear Map of an ellipsoid in $\mathbb{R}^N$ into another ellipsoid in $\mathbb{R}^n$, with $n<N$

Starting from the closed set describing an ellipsoid in $\mathbb{R}^N$: $$\Omega_x = \{ x \in \mathbb{R}^N : (x-x_0)^T\Sigma_x^{-1}(x-x_0) \leq \varepsilon^2 \}$$ where $\Sigma_x \in ...
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0answers
7 views

What's the dimension of the space formed by taking a union of protective lines between two spaces?

I've read that the subspace formed by taking the union of all projective lines that join a point of $\mathbb{P}^{l}$ with a point of $\mathbb{P}^{m}$ has dimension l + m + 1. Why exactly is this the ...
3
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1answer
52 views

Trying to solve exercise $2$ from chapter $5$ of Eisenbud's Geometry of Syzygies book

I am trying to solve exercise $2$ from chapter $5$ of Eisenbud's Geometry of Syzygies book.The problem is as follows: Let $X$ be the union of two disjoint lines in $\mathbb P^3$,or a conic ...
5
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2answers
98 views

Projective bundle of a sum of ample line bundles

Let $X$ be a quasi-projective variety, and let $L_1,...,L_n$ be ample line bundles on $X$. Is that true that if $E= \oplus L_i$, then $$\mathbb{P}(E) \cong Proj(\bigoplus_{i_j \in \mathbb{N}^n} ...
2
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0answers
13 views

Can Hecke Operators be defined on more general spaces of elliptic curves?

Classically, the Hecke Operators act as endomorphisms of $\omega^k_{\mathcal{M}_{ell}(\mathbb{C})}$, defined by noting that there is a distinguished class of covering maps $\widetilde{E}\to E$ given ...
12
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1answer
305 views

Hartshorne Theorem 8.17

I can't understand the proof of theorem 8.17 from Hartshorne's "Algebraic Geometry". Namely, he says that we have an exact sequence $$ 0 \to \mathcal J'/\mathcal J'^2 \to \Omega_{X/k} \otimes ...
3
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1answer
177 views

Theorem 8.17 , Chapter II, Hartshorne

Let X be a nonsingular variety of dim n over an algebraically closed field k. Let Y be an irreducible closed subscheme defined by a sheaf of ideals $\mathscr I$. Then I want to prove that Y is a ...
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1answer
45 views

Derivative of a section of a vector bundle

Let $X$ be a complex algebraic variety and let $E \to X$ be a vector bundle over $X$, with sheaf of sections $\mathcal{E}$. If $s$ is a local section of $\mathcal{E}$, what is the derivative $ds$ ...
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56 views

Dimension of a sheaf cohomology group on a genus 1 curve

Let $\mathcal{M}_{g,1}$ be the moduli space of genus 1 curves with 1 puncture. For simplicity let's take $g > 1$. As usual, there is a natural fibration $C \rightarrow \mathcal{M}_{g,1} \rightarrow ...
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0answers
26 views

How to describe a polynomial relation on $\mathbb{P}(\bigwedge^k V)$, and if the Zariski topology is canonical

I am working with the space $\mathbb{P}(\bigwedge^k V)$, where $V$ is some $n$ dimensional vector space over some field K. In here I want to define a variety, ie a solution to a set of polynomials. ...
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1answer
31 views

Fiber of morphism induced by map on stalks

Given a morphism of schemes $f\colon X\to Y$ and a point $x\in X$, the map on the stalks induces a morphism $\operatorname{Spec}\mathcal{O}_{X,x} \to \operatorname{Spec}\mathcal{O}_{Y,f(x)} $. Is it ...
1
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1answer
24 views

Dimension of the stalk at the point in the closure of an open subscheme

Let $U$ be an open subscheme of $X$ a $\mathbb C$-scheme locally of finite type. I know that $U$ is of dimension $n$ and I have a point $x$ in the closure of $U$. What can be said of the Krull ...
3
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1answer
39 views

Apparently the conormal sheaf of the diagonal constructs differentials … how do I plug in vectors?

Let me specify the construction I am thinking about, then I have a specific question to ask at the end. We are given a morphism of schemes $f: X \to Y$ (let's just say separated to avoid something ...
1
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0answers
18 views

Characterization of Groebner Bases in terms of unicity of remainders

Let $I$ be an ideal of a polynomial ring $k[x_1,\ldots,x_n]$ over a field $k$. A Groebner basis of $I$ is a finite generating set $\{g_1,\ldots,g_m\}$ such that every leading monomial (according to a ...
5
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1answer
49 views

Is this result on the bound of regularity of an ideal true?

I am solving a problem in which i need to use the following result but i am not sure whether the result is true on not: If the ideals $I_0,...,I_n$ are generated by linear polynomials in ...
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0answers
36 views

One of the conditions to be a sheaf [on hold]

Let $A$ a ring and $X=Spec(A)$. Let $B=\left\{D(f)\mid f\in A\right\}$ base open $X$ where $D(f)=X\setminus V(f) $ for $f\in A$ and $V(f)=\left\{p\in X\mid f \in p\right\}$. $O_X(D ...
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1answer
29 views

Regularity and Short Exact Sequence

Suppose $ 0 \to M_1 \to M_2 \to M_3 \to 0$ is a short exact sequence of finitely generated graded $k[x_0,...,x_r]$-modules. Then show that $\mathrm{reg}(M_1) ...
0
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1answer
27 views

Variety of homogenous polynomials which factor as the product of linear forms

Let $V$ be the complex vector space of all homogenous polynomials in three variables of degree $d$ and $\mathbf{P}V$ the corresponding projective space. Let $X$ be the subset of all those homogenous ...
35
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3answers
3k views

What use is the Yoneda lemma?

Although I know very little category theory, I really do find it a pretty branch of mathematics and consider it quite useful, especially when it comes to laying down definitions and unifying diverse ...
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0answers
32 views

Trying to Compute Regularity and degree

Definition: For a finite subset $X \subset \mathbb P^r$,the Hilbert function $H_X(d)$ is constant for large $d$ and its value is the number of points in X,usually called the degree of $X$. Let ...
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1answer
26 views

Torus action and multigrading.

Let $G$ be an algebraic group and $T$ the maximal torus. Suppose that $T$ acts on $G$. Do we have a multigrading on $\mathbb{C}[G]$? How to define the multigrading corresponding to the $T$-action? ...
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0answers
41 views

Does Nagata theorem hold in a field that is not algebraically closed?

Let $R$ be a finitely generated $k$ - algebra and $G$ be a reductive group acting rationally on it. Then a theorem of Nagata says that the invariant ring $R^G$ is also finitely generated. Here $k$ ...
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2answers
90 views

$V=\mathcal{Z}(xy-z) \cong \mathbb{A}^2$.

This question is typically seen in the beginning of a commutative algebra course or algebraic geometry course. Let $V = \mathcal{Z}(xy-z) \subset \mathbb{A}^3$. Here $\mathcal{Z}$ is the zero locus. ...
1
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1answer
34 views

Problem in understanding the proof of lemma $7.2.5$ in Liu's book

Let's analyze the proof of the following lemma: Lemma: Let $X$ be an integral, Noetherian scheme and let $f\in K(X)^\ast$, then for all but finitely many points $x\in X$ of codimension $1$ we have ...
2
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0answers
23 views

Where does the linear map whose minors determine the rational normal cone, come from?

$\newcommand{\C}{\mathbb{C}}$ $\newcommand{\im}{\operatorname{Im}}$ Let the rational normal cone be given by the image of $\Phi: \C^2 \to \C^{d+1}$, parametrized by $(s,t) \to (s^d, s^{d-1}t,...t^d)$. ...
2
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1answer
54 views

Center of a semisimple group and irreducible representations

Suppose that I am over an algebraically closed field of char $0$, and $G$ is a simply connected semisimple group. For a dominant weight $\lambda$, there is an irreducible representation ...
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0answers
29 views

A basis for forms of degree $d$ (Fulton, 2.35)

I am trying to solve this exercise from Fulton's book: (2.35)(c) Let $L_1, L_2, \dots,$ and $M_1, M_2, \dots$ be sequences of nonzero linear forms in $k[X,Y]$ and assume no $L_i = \lambda M_j$ for ...
9
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1answer
240 views

Automorphism group of the configuration of lines on a cubic surface and quadratic transformations

It's well known that the automorphism group of the configuration of 27 lines on a smooth cubic surface in $\mathbb{P}^3$ (over a field containing all 27 lines) is isomorphic to the Coxeter group of ...
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0answers
28 views

Implicit equation that can not be parameterized. [on hold]

Is there an example of an implicit equation that can not be parameterized ?
15
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0answers
193 views

Complex manifold with subvarieties but no submanifolds

Note, I have now asked this question on MathOverflow. There are examples of compact complex manifolds with no positive-dimensional compact complex submanifolds. For example, generic tori of ...
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0answers
25 views

How can I “groupify” an elliptic curve over a non-field?

The book Primes of the form $x^2+ny^2$ by David A. Cox contains the following definitions regarding an elliptic curve (by which he means an equation $y^2=4x^3-g_2x-g_3$ such that ...
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0answers
59 views
+100

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 ...
1
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1answer
22 views

What are the stalks of $\Gamma_Z(F)$ for a locally closed subset $Z$ and a flabby sheaf $F$

My references for the following notations are Iversen & Hotta,Takeuchi, Tanisaki. Let $Z \subset X$ be a locally closed set, and $F \in Sh(X)$. Let $Z \subset W \subset X$ be an open subset of ...
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1answer
21 views

What is affine line in Ext(F'',F')?

Let X be a projective scheme over a field k , and $0\rightarrow F'\rightarrow F \rightarrow F''\rightarrow 0$ be a short exact sequence of semi-stable sheaves with same reduced Hilbert polynomial. ...
2
votes
2answers
267 views

Finite subset of projective $n$ space is a variety

How do I prove that a finite subset $M$ of the $n$-dimensional projective space is a variety? I tried finding a homogeneous polynomial of degree one that vanishes at an arbitrary x in M and then ...
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0answers
31 views

Is this toric variety the blowup of $\mathbb C^2$ at some point?

Let $u_1=e_1,\quad u_0=e_1+2e_2,\quad u_2=e_2$. Consider the fan consisting of the following cones $\sigma_1= \langle u_1,u_0\rangle$, $\sigma_2=\langle u_0,u_2\rangle$ and their faces. Then the toric ...
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0answers
50 views

Equivalence of smoothness and freeness of sheaf of differentials

Let $S$ be a regular locally Noetherian connected scheme, $f:X \to S$ a morphism of finite type with $X$ irreducible. Let $x \in X$ and $s = f(x)$ such that $$ \dim \mathcal{O}_{X,x} = \dim ...
2
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1answer
41 views

Homogeneous spaces of elliptic curve

Let $d>1$ be a cube free natural number and $a,b,c$ are natural numbers greater than 1 with $abc=d$. How to explain that the curve $D: ax^3+by^3+cz^3=0$ is homogeneous space (over $\mathbb{Q}$) ...
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0answers
24 views

On Kapranov's “On the derived categories of coherent sheaves on some homogeneous spaces”

As a graduate master student I am reading Kapranov's paper "On the derived categories of coherent sheaves on some homogeneous spaces" (1988). One problem is that the paper assume lot of notations and ...
0
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1answer
51 views

Ellipse's farthest point to another point

I am trying to find the farthest and closest points of a ellipse without using any brute force type of coding. The processing power is limited so it should be as pinpoint as possible. I have tried a ...
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0answers
59 views

Unramified at a point $x \in X$ if and only if $\Omega _{X,x} = 0$

This is Corollary 6.2.3 in Liu's book. Let $f: X \to S$ be a morphism of finite type of locally Noetherian schemes. Then $f$ is unramified at a point $x \in X$ if and only if $\Omega_{X/S, x} = ...
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3answers
73 views

Scheme over a not algebraically closed field

I'm trying to make a scheme out of an affine variety $V\subset k^n$, when the base field $k$ is not algebraically closed. I wonder if $V$ is the spectrum of its ring of regular functions ...
4
votes
1answer
40 views

Smooth curve of genus $1$ in $\mathbb{P}_{\mathbb{C}}^1\times \mathbb{P}_{\mathbb{C}}^1$.

This question comes from Gathmann's notes of Algebraic Geometry: Show that $$\{((x_0:x_1),(y_0:y_1)): (x_0^2+x_1^2)(y_0^2+y_1^2)=x_0x_1y_0y_1\}\subset \mathbb{P}_{\mathbb{C}}^1\times ...
0
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1answer
36 views

Is there any large diffeomorphisms of $S^{n}\times S^1 $like Torus?

We know that a Torus is mapped onto itself in a special discontinuous transformation given by $PSL(2,\mathbb{Z})$. Thinking of torus as $S^{1}\times S^{1}$ and thus as a lattice, we can easily show ...
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0answers
20 views

Hilbert function under base change

I need help to understand how Hilbert functions behave under base change. Is the Hilbert function of a scheme $X/k$ equal to that of $X/\bar{k}$? It seems to me the answer should be no but I can't ...
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0answers
26 views

intersection of a line and a curve [on hold]

I have this question let $3x+4y=100$ find $x,y\in$ $\mathcal {} \mathbb{Z}$ such that $y^2+x^2=c $ is the smallest. using calculus I can find that $x=12, y=16$. But this is the only solution it ...
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0answers
34 views

Does this theorem hold for real part of a toric variety? + Reference request - Real toric varieties.

Let $X$ be a complex toric 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 ...
1
vote
1answer
37 views

Application of Bertini's theorem

I am suffering with this rudimentary question related with Bertini's theorem. As one can see, in Hartshorne's book 'Algebraic Geomery V.1.2', there is one application of Bertini's theorem. I am really ...