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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79 views

The $2 \times 3$ matrices with rank $\leq 1$ cannot be defined by two polynomial equations

Let $X$ be the space of all ${2 \times 3}$ matrices over $\mathbb{C}$ that have rank at most 1. This is naturally a subspace of $\mathbb{C}^6.$ We can express $X$ using 3 polynomial equations, namely ...
4
votes
1answer
50 views

Eisenbud-Harris Exercise II-14, limit scheme isomorphic to triple point and remembers both tangent line, osculating $2$-plane to subscheme

Let $C$ be the subscheme of $\mathbb{A}_K^n$ given by the ideal$$J = (x_2 - x_1^2, x_3 - x_1^3, \ldots).$$A closed point in $C$ is of the form $f(t) = (t, t^2, t^3, \ldots, t^n)$, for $t \in K$; that ...
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0answers
21 views

intersection between conic and cubic

Let $ A=V(\alpha)$ a non-singular conic in $\mathbb{P}^2(\mathbb{C})$, $V(\delta_1)$ and $V(\delta_2)$ two cubic such that : $D_1$ and $D_2$ meet $A$ tangentially at six distincts points $P_1,...,P_6$ ...
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0answers
77 views

Global sections and divisors

I'm trying to understand the proof of the Theorem at page 163 from Mumford, Abelian Varieties, and I have a question about one step. This is the situation: $X$ is an abelian variety (hence there's ...
0
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1answer
98 views

Strict transform of blow up

Let $X$ be a smooth projective variety over $\mathbb{C}$. Consider the blow up of $X$ about a closed subvariety $Z$. Let $X'=Bl_Z(X)$. Let $Y$ be a smooth irreducible divisor of $X$ properly ...
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0answers
64 views

Why is the intersection of algebraic subsets of an algebraic variety again an algebraic subset?

This question is inspired by the Wikipedia article on the Zariski topology: https://en.wikipedia.org/wiki/Zariski_topology Since I know next to nothing about algebraic geometry, and no advanced ...
0
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1answer
44 views

Determining the projective closure of a variety.

Consider the twisted curve in $\Bbb{C}^3$ denoted by the ideal $\langle x^2-y, z-xy\rangle$ I did it by homogenizing the generators: I got $\langle x^2-yw, zw-xy\rangle$. The projective closure ...
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1answer
35 views

Isogenic elliptic curves. Number of points and zeta function

Is there any book or other reference where I can find a complete proof of the following fact? If $E$ and $E'$ are two isogenic curves (over $\mathbb{F}_q$, where $q=p^a$), then for any $n \ge 1$ the ...
0
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1answer
52 views

Definition of pullback of a Weil divisor on an abelian variety?

We are on an abelian variety, so Cartier divisors, Line bundles and Weil divisors are all equivalent. I would like to see the pullback of a Weil divisor. Is it true that, if $D=\sum n_i E_i$, then the ...
0
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2answers
51 views

restrictions of rational sections of an invertible sheaf.

Let $(X,\mathscr O_X)$ be an integral scheme with function field $K(X)$ and let $\mathscr L$ be an invertible sheaf on $X$. Moreover suppose that $\{U_i\}$ is an open covering of $X$ such that $\...
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182 views

Is there a universal statement for the construction of global Proj?

Let $X$ be a scheme and $\mathcal{A}$ be a sheaf of $\mathbb{Z}_{\ge 0}$-graded $\mathcal{O}_X$-algebras. From the data above there is a construction which gives the "global Proj" $Proj \mathcal{ A} ...
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0answers
16 views

local representation of “logarithmic connection”

Let X be a Riemann compact surface, $D\subset X$ be a finite subset, and (E,$\nabla$) be a logarithmic connection. And let $z$ be a local coordinate at $p\in D$, why $\nabla $ can be written by: $\...
1
vote
1answer
51 views

Zariski closure of $T:= \{(t_1,t_2,t_3,t_1t_2t_3^{-1})|t_i\in \mathbb{C}^*\}\subseteq \mathbb{C}^4$?

Let $V= \mathcal{V}(\langle xy-zw\rangle)\subseteq \mathbb{C}^4$ be an affine variety. The set $T:= \{(t_1,t_2,t_3,t_1t_2t_3^{-1})|t_i\in \mathbb{C}^*\}$ is a torus contained in $V$. I am trying to ...
2
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1answer
62 views

Given a locally ringed space there is a bijection between open and closed sets of $X$ and idempotent elements of $\mathcal{O}_X(X) $

This is a problem from Gortz and it does NOT assume that the underlying space is the spectrum of the ring or anything like that. Now I proved easily that given a clopen set of $X$ there is an ...
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0answers
54 views

Roots of canonical line bundles that are not necessarily square roots

I understand that holomorphic square roots of the canonical line bundle of a compact Riemann surface always exist, and that there are $2^{2g}$ choices of such a root. But what about further roots? ...
0
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1answer
29 views

The isotropy of the action of $SU(3)$ on $\mathbb CP^2$

Consider the action of $SU(3)$ on the complex projective plane $\mathbb CP^2$. How we can find the isotropy group?
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0answers
39 views

A complement of Milnor's book on singularities with exercises/examples

I'm currently reading "Singular Point of Complex Hypersurfaces" by Milnor. This is really a great book, but I did realize I didn't saw any concrete examples, computations etc ... I was wondering if ...
4
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0answers
43 views

Obtaining a nice map to a curve by using blowups

Let $X$ be a smooth and projective variety over a finite field (separated, finite type, integral). Then after performing a number of blowups I should be able to find a proper surjective map from $X$ ...
3
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4answers
186 views

Diagonal morphism of regular variety is a regular embedding

Let $X$ be a regular $k-$variety (i.e. all of its local rings are regular) of pure dimension $d$. Then I would like to show that the diagonal morphism $X\rightarrow X\times_k X$ is a regular embedding ...
3
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0answers
100 views

Is this true that, any algebraic curve has finitely many singularities?

Can we say that any algebraic curve has finitely many singularities?
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0answers
45 views

Covering maps of schemes.

A curve $X$ is modular if there is a finite covering $X_0(N)\rightarrow X$. What does covering mean in this context, and for more general morphisms of schemes? Just covering as topological spaces?
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2answers
54 views

Scheme theoretic 'class inclusions'

For lack of a better name*, I call the following two things class inclusions: $$1) \quad\textbf{Magma}\supset \textbf{Semigroup}\supset\textbf{Monoid}\supset \textbf{Group}\supset \textbf{Abelian ...
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27 views

Cohomology of the Munford line budle on an Abelian variety

Let $X$ be an Abelian variety over a field $k$; $L$ line bundle on $X$. I would like to calculate the cohomology of the Mumford line bundle $\Lambda(L)=m^*L\otimes p_1^*L^{-1}\otimes p_2^*L^{-1}$; ...
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0answers
30 views

Explain Betti diagram of a minimal free resolution for a simplical complex

I am self-learning algebraic geometry and reading the book The Geometry of Syzygies A Second Course in Algebraic Geometry and Commutative Algebra and I want to ...
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1answer
53 views

The isotropy of the complex projective plane for the action of $SU(3)$

If we consider the action of the compact real form $SU(3)$ of $SL(3,\mathbb C)$ on the space $\mathbb C^3$. Since the action is transitive, how to find the stabilizer $G_x$? Is it useful to find ...
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0answers
64 views

Riemann-Roch and quartic

I know very little in algebraic geometry, but I want to learn!! So I know the Riemann-Roch theorem as follow: let $$L(D)=\{\text{ meromorphic functions, s.t. }\operatorname{div}(f)\geq D \}$$ and $$...
5
votes
1answer
86 views

Hilbert polynomial for a dimension zero projective variety by taking an affine chart

I am looking at exercise 12.21 from Gathmann's notes on algebraic geometry. I am given a homogeneous ideal $$I \unlhd k[x, y, z] $$ with a dimension $0$ projective locus. WLOG, we assume that this ...
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0answers
46 views

Exterior Algebra VS Torsion

Let $C$ be an irreducible and reduced rational curve, and $f: \mathbb P^1\rightarrow C$ be the normalization. If $\mathcal F$ is a coherent sheaf of rank $r$ over $C$, then I was wondering if we can ...
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1answer
33 views

To solve large systems of multivariate polynomial equations

Nicolas Courtois et al. proposed the eXtended Linearization(XL) method to solve the systems of multivariate polynomial equations and analyzed the time complexity. Polynomial when the number of (...
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68 views

Vanishing of Cohomology of Affine Schemes — Proof

I was following Ravi Vakil notes FOAG (http://math.stanford.edu/~vakil/216blog/FOAGjun1113public.pdf) and I cannot understand Th.18.2.4, at least not all of it. What I can follow is the first part: ...
3
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0answers
103 views

Understanding an application of Riemann-Roch in an article

I saw the following in an article: Let $C$ be an irreducible smooth projective curve over an algebraically closed field $K$ and let $g$ be its genus. By Riemann-Roch, if N is large enough for every ...
2
votes
1answer
40 views

Support of quotient sheaf of ideal sheaves with same support

I'm not very sure about this argument. Let $\mathscr{I},\mathscr{J}$ two ideal sheaves (you can think about ideal sheaves over a projective variety or even the projective space itself) and assume that ...
3
votes
1answer
43 views

The Mumford line bundle of $(-1)^* L$

Let $X$ be an abelian variety over a field $k$, $L$ a line bundle on $X$. Let $\varphi_L : X \to X^t$ be the morphism obtained by considering the Mumford line bundle $\Lambda (L) = m^*L \otimes p_1 ^...
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0answers
60 views

Sheaf hom of coherent sheaves

Let $(X,\mathscr O)$ be a locally ringed space, and let $\mathscr F$, $\mathscr G$ be $\mathscr O$-modules. Consider the following facts: $\newcommand{\sF}{\mathscr F}\newcommand{\sG}{\mathscr G}\...
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0answers
43 views

Property of a mapping on the tensor product

Let $A$ and $B$ be rings and consider the tensor product $A \otimes_R B$. Hartshorne writes that: To give a homomorphism $A \otimes_R B$ into a ring is the same as to give a homomorphism of $A$ and $...
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2answers
37 views

A question about a projective map in Karen Smith's “An Invitation to Algebraic Geometry” [closed]

The following example is taken from Karen Smith's book "An Invitation to Algebraic Geometry", pg. 45. Let $C=\Bbb{V}(zx-y^2)\subseteq \Bbb{P}^2$ be a plane conic. This is a projective variety. ...
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2answers
42 views

Question regarding proof in Hartshorne on existence of the fibered product

Let $X$ and $Y$ be schemes over $S$. For the time being let us assume that $X$, $Y$, and $S$ are affine schemes such that $X= Spec \hspace{0.5mm} A$, $Y = Spec \hspace{0.5mm} B$, and $S= Spec \hspace{...
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votes
1answer
39 views

Constraining a choice of vector from a given space

I have a hyperplane equation: $$\vec{x}\cdot \vec{w} = 0$$ Now, on its own, this just means that $\vec{x} \in Nullspace(\vec{w})$. I have additional constraints on the components of $\vec{x}$, ...
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0answers
51 views

Why is $Supp D=Supp(\pi^*(\pi(Supp D)))$?

I'm trying to understand the proof of the theorem at page 163 of Mumford, Abelian Varieties. At some point we have the following situation: $X$ is an abelian variety, $D$ is an effective Weil divisor ...
1
vote
0answers
53 views

Seesaw principle

Let $k$ be an algebraically closed field of characteristic 0. The seesaw principle in algebraic geometry usually goes like this: let $T$ be a complete variety, let $X$ be an integral scheme of finite ...
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1answer
55 views

Morphism of finite type schemes not surjective. Is there a closed point in the complement of the image?

I have the following proplem: Let $k$ be an algebraically closed field and let $X,Y$ be schemes of finte type over $k$. Now let $f:X\to Y$ be a morphism of schemes that is not surjective. Question: ...
3
votes
1answer
87 views

Classification of subschemes of $\mathbb{A}_K^2$ of dim $0$, deg $4$, $5$ with support at origin

What is the classification up to isomorphism of subschemes of $\mathbb{A}_K^2$ of dimenion $0$ and degrees $4$ and $5$ with support at the origin? Which are isomorphic as schemes over $\text{Spec}\,K$?...
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votes
0answers
122 views

Legitimacy of drawing a complex curve like a plane curve

In algebraic geometry, we often consider a complex algebraic curve, and in order to get some intuition, we often draw it on the plane as if it were a plane curve. In most cases it turns out that the ...
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0answers
79 views

Need a counterexample to show that Cl$(X\times Y)$ is not always same as Cl$(X) \oplus $Cl$(Y)$

Recall that for a quasi projective variety $X$ one can define the Divisor Class Group denoted by $\operatorname{Cl}(X)$. Suppose $X$ and $Y$ be two quasi projective varieties.What is the ...
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0answers
34 views

Examples of the primitive decomposition of a form

Let $(X,\omega)$ be a Kahler manifold of dimension $n$, let $L = \omega \wedge -$ and let $\Lambda$ denote its adjoint. There is a unique ''primitive decomposition'' of a $k$-form $u$ which looks like ...
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0answers
39 views

Riemann-Roch Space for Quotient Curve

Let $C$ be a curve defined over a finite field $\mathbb{F}_q$. Let $\{f_1,..f_m\}$ be a basis for the riemann-roch space of functions, L(D), for the divisor $D= t\infty$. Suppose you have a subgroup ...
5
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0answers
82 views

Intersection of a quadric and cubic in $\mathbb{P}^3$

My question is drawn from Miles Reid's textbook Undergraduate Algebraic Geometry, p. 116. Let $S \subset \mathbb{P}^3$ be a smooth, irreducible cubic. Let $l_1, l_2, l_3, l_4 \subset S$ be ...
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votes
1answer
55 views

Algebro-geometric proof of Cayley Hamilton theorem

I am looking for a reference on the algebro-geometric proof of C.H. Theorem: Every square matrix satisfies its characteristic polynomial. There are several points I don`t understand reading my ...
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0answers
31 views

Basic ideals exercise, and a question about notation definition

From this this book. Given a finite set $\left\{f_1,f_2,\ldots,f_r\right\} \subset R$, the ideal $I$ generated by this set is denoted $f_1, f_2, \ldots , f_r$ and consists of all the sums $f_1h_1 ...
3
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0answers
42 views

Chern classes of the associated vector bundle of a branched covering

Let $f \colon X \to \mathcal{Q}_7$ be a branched covering of degree $3$ of a $7$-dimensional smooth projective quadric $\mathcal{Q}_7 \subset \mathbb{P}_8$, where $X$ is a smooth connected projective ...