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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14 views

Calculate rotation matrix starting with 8 coordinates from a box to an axis-aligned box

I've got a rectangular box that's described by the coordinates of its 8 corners. Now I want to calculate the rotation matrix which would rotate this box so its edges align with the coordinate system. ...
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30 views

How can I blow-up a smooth projective surface with certain conditions?

Let $X$ be a smooth projective surface and $K$ a canonical divisor on $X$. Suppose $V$ and $W$ are subspaces of $H^0(X, \mathcal{O}_X(nK))$ (for $n$ large). Q: How can we blow-up $X$ to obtain $\pi: ...
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1answer
49 views

Degree of a projective variety

Let $X \subset \mathbb{P}^n$ be a projective variety of dimension $k <n$. By an equivalent definition of dimension, $k$ is the smallest integer such that there exists an open set of $G(n-k-1,n)$, ...
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41 views

Quasicoherent-sheaves and pushfoward

How to prove the proposition in the picture below? It seems to be easy, but I am a beginner. Thanks in advanced for your help!
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28 views

Question on the matrix of a Kaehler Metric in Normal Coordinates

I am currently studying normal coordinates on a Kaehler manifolds: Let $h$ be a Kaehler metric on a complex manifold $M$ and let $p \in M$. Let $(z_1,..,z_n)$ be a coordinate chart such that $h$ is a ...
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1answer
55 views

degree of an etale cover of the affine line

Let $X\subset \mathbb{A}^N_k$ be an irreducible smooth variety over an algebraically closed field $k$. Suppose we have an etale map $\pi:X\to \mathbb{A}^1_k$. Are there any bounds on the degree of ...
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0answers
75 views

Can someone give me the spherical equation for a 26 point star?

This is the object that I am trying to find the volume of. This can be treated as a "26 point star". What I need is an equation to describe it. If anyone has that surface in spherical ...
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1answer
30 views

Two questions about Schubert calculus and Schur functions.

I am reading the file. I have a question on pae 28. How to prove that $[X_{\{2,4\}}] = S_{(1)} = x_1 + x_2 + \cdots$ and $S_{(1)}^4 = 2 S_{(2,2)} + S_{(3,1)} + S_{(2,1,1)}$? I tried to verify ...
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34 views

Schubert calculus and number of lines satisfying some properties.

I am reading the file. I have a question on pae 18. It is said that: Given a line in $\mathbb{R}^3$, the family of lines intersecting it can be interpreted in $G(2, 4)$ as the Schubert variety $$ ...
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1answer
53 views

Can one have a nontrivial 'resolution of singularities' of a smooth variety?

Suppose $z_1,z_2$ are coordinates on $\mathbb{A}^2$ and $(w_1,w_2)$ homogeneous coordinates on $\mathbb{P}^1$. We can define a subvariety $X \subset \mathbb{A}^2 \times \mathbb{P}^1$ by $w_1z_2 - ...
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37 views

Detail regarding tangent spaces and dual varieties from Harris's Algebraic Geometry: A First Course

In Harris's Algebraic Geometry: A First Course, Example 16.20, the author shows that the dual of the dual variety $X^{*}$ is the original variety $X$. I think in chapter 15, Harris mentions that he'll ...
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1answer
44 views

Question about Normal Coordinates on a Kaehler Manifold

I am currently reading FY Zheng's textbook, "Complex Differential Geometry". In section 7.4 Proposition 7.14, he is trying to prove thata metric $h$ Kaehler is equivalent to the statement, "For any $p ...
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1answer
59 views

Non-linear equivariant maps between group representations

Given two representations $\pi_1$ and $\pi_2$ of a group $G$ (let's say it's a compact Lie group), a natural thing to study are linear equivariant maps A between them: $$ A \pi_1 = \pi_2 A $$ I'm ...
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95 views

Can you integrate on a scheme?

As the question suggests, can you integrate on a scheme? How? I don't even know if this is even a well-posed question...
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30 views

Re-write the Riemann Roch theorem

How can I write the Riemann-Roch Theorem with the following definitions: Definition $1$: $\mathcal{F}$ is an scheme of fractional ideals on $C$ if is coherent and for all $P\in C$, the stalk ...
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0answers
26 views

How do I define a Gorenstein curve?

How can I define a Gorestein curve with the following definition: For each $\lambda\in\Omega_{k(C)/k}$, define $\omega_{\lambda}\in {\rm Frac}(C)$, such that for all $P\in C$ the stalk ...
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3answers
121 views

When is a Morphism between Curves a Galois Extension of Function Fields

My apologies if this question has already been answered somewhere on this site: when I searched, I could only find specific examples rather than the general question. It is known that the category of ...
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34 views

$\mathbb{Q}$-rational point of a moduli space and being defined over $\mathbb{Q}$

Mazur's theorem on the torsion points of an elliptic curve includes the fact that each possibility which can occur does in fact occur for infinitely many rational elliptic curves. I've read that this ...
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1answer
48 views

Lifting of partial section

Does someone know if the following is true, and if so could you provide a reference. Let $X$ be a smooth $S$-scheme, let $Z$ be a closed subscheme of $S$, and let $t : Z \to X$ be an $S$-morphism. ...
2
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0answers
21 views

Existence of real structure on CY m-fold

Suppose $M$ is Calabi-Yau $m$-fold with complex structure $J$, Kahler form $\omega$, metric $g$ and holomorphic $m$-form $\Omega$. What are the conditions on $M$ for the existence of a map $\sigma: M ...
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32 views

The trace ideal of a non zero $R$-module

Let $R$ be a commutative ring with identity and $M$ be a cyclic $R$-module, we may define the ideal $tr(M)$ associated with $M$, the sum of the ideals $f(M)$, for all $R$-homomorphisms $f \in ...
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74 views

A Geometric Description of Injective Modules

I've found that when studying commutative algebra, thinking of things in terms of their algebro-geometric interpretation helps them stick as well as motivates otherwise odd and abstract concepts. ...
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55 views

Quick question: Determinant bundle is Cartier?

$X$ is an algebraic surface (i.e. compact complex, which embeds in a projective space). $V$ is a vector bundle of rank 2 over $X$. Why is $\det{V}=\mathcal{O}_X(D)$ for some divisor $D$? Is there a ...
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34 views

Secant variety and tangent lines (Harris, Algebraic Geometry: A First Course)

Given a (smooth) projective variety $X\subset \mathbb{P}^n$, we can define a rational map $s:X\times X\rightarrow G(1,n)$ that takes a pair $(p,q)\in (X\times X)\setminus \Delta$ not on the diagonal ...
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1answer
87 views

difference between $\mathbb{A}^n$ and $k^n$ [closed]

Can any one please give me actual definition of $\mathbb{A}^n$ ? Some books says its same as $k^n$ but Eisenbud says if $k$ is not algebraically closed both are different
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1answer
58 views

Help with my proof of maximum number of intersection points of n lines being n(n-1)/2

I know that this question is a hoary old chestnut, but I have never seen a proof before working one out myself, so I'd like you to help me see if mine is rigorous enough. Obviously with $n$ linear ...
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0answers
48 views

Is functional analysis related to or used in algebraic geometry in any way?

I'm curious about whether there's a link (and, no, this question was not motivated by the fact that Grothendieck used to be a functional analyst!) between these two subjects. Are the techniques from ...
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0answers
23 views

Restriction of hypersurfaces to canonically embedded curve is surjective?

Let C be a canonically embedded curve (irreducible, projective, smooth, over an algebraically closed field) of genus g at least 3, in particular not hyperelliptic. Is the restriction map of degree d ...
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1answer
18 views

Equation of parabola from 2 points and an angle at the first point

A parabola starts at a coordinate A and ends at coordinate B. Angel of tangent through A is given 'theta'. With these data (A,B,theta) how can I get an equation of a parabola?
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25 views

structures on a vector space

In The Hodge theory of Soergel bimodules, $(W,S)$ denotes a Coxeter system, and for simple reflections $s,t\in S$ denote by $m_{st}$ the order of $st$. Fix a finite dimensional real vector space ...
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1answer
22 views

Generic fiber of a scheme over a DVR

What is (usually?) meant by the generic fiber of a scheme over a discrete valuation ring? I've seen in some talks now, could somebody give a precise definition? Thank you very much in advance!
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1answer
29 views

Find the invariant polynomial space of a finite matrix group.

This is an exercise from Ideals, Varieties and Algorithms by Cox et al. Denote the finite matrix group $\{\pm I_2\}\subset GL(2,k)$ by $C_2$. It is known that the invariant polynomials under $C_2$ ...
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1answer
54 views

Classical and Modern Zariski Topologies

I am trying to understand the connection between the "classical" Zariski topology on a variety, that is closed sets defined as the vanishing set of some ideal, $V(A)$, and the "modern" topology on the ...
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34 views

application of local fiber dimension theorem

Let $\pi : X \rightarrow Y$ be any morphism of projective varieties, $Z$ locally closed subset in $X$ and $p \in Y$ a general point. Denote by $Z_p$ the fiber $X_p=\pi^{-1}(p)$ restricted on $Z$. The ...
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1answer
43 views

How to calculate the canonical divisor of projective space bundle?

Suppose $X$ is a smooth variety, $E$ a rank $r$ vector bundle on $X$, $E^*$ its dual bundle, how do we show the canonical divisor can be expresses as $$K_{\mathbf{P}(E^*)}\cong \pi^*K_{X}\otimes ...
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50 views

The induced map on stalks is well-defined

$\require{AMScd}$ Let $\phi:\mathscr{F\to G}$ be a morphism of sheaves on $X$, let $\mathscr F_P$ be a stalk of $\mathscr F$ at $P\in X$, and let the stalk map $\phi_P:\mathscr F_P\to\mathscr G_P$ ...
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2answers
33 views

How many ways to compute a polynomial expression?

Hi guys is there an algorithm or procedure that could tell me how many and what are the ways to compute an algebraic expression? Just to make a stupid example... $$x + y + z = (x + y) + z = x + (y + ...
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0answers
16 views

Sections to projection morphisms

Let $R$ be a complete DVR, $X, Y$ schemes over $\mathrm{Spec}(R)$. Denote by $f:X \times_R Y \to X$ be a natural projection morphism. Does the morphism $f$ has a section i.e., a ...
3
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2answers
73 views

Why is $GL(n, \Bbb R)$ open in the Zariski topology?

The general linear group $GL(n,\mathbb{R})$ is said to have Zariski topology, but I do not understand how. Can anyone explain this to me?
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2answers
59 views

Closed sets in Spec(k[X,Y])

On page 74 of Mumford's red book (attached) it is stated that a proper closed set in Spec(k[X,Y]) is composed of finitely many irreducible curves and finitely many closed points. Why does such a union ...
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47 views

Chern class of $\mathbb C\mathbb P^2 \# 6\overline{\mathbb C\mathbb P^2}$

I would like to compute the first Chern class of $\mathbb C\mathbb P^2 \# 6\overline{\mathbb C\mathbb P^2}$ in terms of the generators of $\mathbb C\mathbb P^2$ and $\overline{\mathbb C\mathbb P^2}$. ...
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1answer
45 views

What does scheme with an action mean?

What is the meaning of the following quote? "A graded Artin A-algebra over a field is the same as an artinian scheme with an action of the multiplicative group. " My main problem is understanding ...
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25 views

How to prove the surjectivity of Hartshorne Exercise II.2.15(c)

The exercise says: $$Hom_{\mathcal{Var/k}}(V,W) \longrightarrow Hom_{\mathcal{Sch/k}}(t(V),t(W)) $$ is a bijection. I don't know how to prove the surjective part.
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1answer
56 views

The “Hartshornian” sheafification of a sheaf

Given a sheaf $\mathscr F$ on $X$, how does one show that its sheafification (in the sense of Hartshorne) is isomorphic to it? The most obvious thing to do is a universal property argument: since ...
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45 views

Finite colength ideals in a power series ring

$\newcommand{\Hilb}{\operatorname{Hilb}}$Let $I\subset R:=\mathbb{C}[[x,y]]$ be an ideal. Then, $I$ is said to be of colength $n$ if $\dim_\mathbb{C}(R/I) = n$. For example the ideal $(x,y)$ has ...
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0answers
30 views

Convex hull of curve

Let $f_1, \ldots, f_n \in \mathbb{R}[t]$ and let $S = \{(f_1(t), \ldots, f_n(t)) : t \in \mathbb{R}\}$. Is there a method of computing the convex hull of this set? How about if each $f_i$ has the same ...
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0answers
35 views

What is the sheaf of endomorphisms?

I saw this term or symbol “End$(\mathcal{F})$", where $\mathcal{F}$ is a quasi-coherent sheaf, in some places, e.g. First Ext group of a sheaf. But I found nothing on Google when I typing "sheaf of ...
2
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1answer
49 views

How to show that $I(V(f))=\left\langle f\right\rangle$ in $\mathbb{R}$.

I encountered several similar questions in Ideal, Varieties and Algorithms by Cox et al. I showed that $I(V(xyz))=\left\langle xyz\right\rangle$ by arguing that if $f$ vanishes whenever $x=0$, then ...
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0answers
22 views

Locating a point in 2D using only differences between distances

Let $R_1,R_2,R_3,R_4,T$ be points on a 2D plane, as in this figure. $R_1,R_2,R_3,R_4$ are reference points with known positions. The goal is to find the position of $T$. $d$ is the Euclidean ...
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24 views

Sections of families of curves

Let $R$ be a complete discrete valuation ring with fraction field, say $K$, $\pi:X \to \mathrm{Spec}(R)$ be a flat, projective, surjective morphism, where $X$ is a regular scheme of dimension $2$. ...