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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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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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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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 ...
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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: ...
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1answer
89 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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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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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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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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40 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 ...
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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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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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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 ...
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43 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 ...
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2answers
54 views

Asymptotes of $(x(t),y(t)) = \bigg(\frac{1+t^2}{2+t^3}, \frac{t}{2+t^3}\bigg)$, collinear points, …

Consider the curve: \begin{equation} (x(t),y(t)) = \bigg(\frac{1+t^2}{2+t^3}, \frac{t}{2+t^3}\bigg) \end{equation} Question 1: What are his asymptotes? Answer: In projective space: $[(2+t^3,1+t^2,t)]$...
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27 views

Ample, Very ample line bundles on projective space

I have been reading Hartshorne. I am trying to understand a bit about a0mple, very ample line bundles, line bundles generated by global sections. I am trying to find these in case of the Projective ...
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2answers
26 views

Proving that the spectrum of a ring generates a topology

I am trying to understand the spectrum topology. In this topology, for any ideal $I\subset R$ (where $R$ is a commutative ring), a closed set $\Bbb{V}(I)$ is defined to be all prime ideals contained ...
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1answer
120 views

What's the difference between $\mathbb{A}^n$ and $\mathbb{A}^{n+1}$?

Besides the obvious difference in topological dimension. If you want to distinguish between $\mathbb{R}$ and $\mathbb{R}^2$, take an open set in the plane, remove a point, then it's still connected. ...
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1answer
48 views

Exact Sequence of Line Bundles on $\mathbb{P}^{2}$

I'm considering an example in the great book "Mirror Symmetry" where they consider the exact sequence of line bundles $\mathcal{O}(-2) \to \mathcal{O}(-1) \oplus \mathcal{O}(-1) \to \mathcal{O}$, ...
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26 views

Griffiths and Harris $\mu=\mathcal{H}(\mu)+dd^*G(\mu)$

Griffith and Harris state on page $116$ that for a closed form $\mu$ on a Kahler manifold of type $(p,q)$ we have $$\mu=\mathcal{H}(\mu)+dd^*G(\mu)$$ Here $$\mathcal{H}:\Omega^{p,q}(M)\to\mathcal{H}^{...
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How to find the distinguished points of this cone?

Consider the cone - $\langle e_1,-e_1+2e_2\rangle$ in the lattice $N=\mathbb Z^2$. Then it has the following faces - $0=\{(0,0)\}$ $\rho_1=\langle e_1\rangle$ $\rho_2=\langle-e_1+2e_2\rangle$ $\...
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34 views

Vanishing set of a pullback section

Let $f\colon X\to Y$ be a morphism of schemes and let $\mathcal{F}$ be an $\mathcal{O}_Y$-module. Let $s\in H^0(Y,\mathcal{F})$. If I'm not mistaken, then $(f^{-1}s)_x=s_{f(x)}$ for all $x\in X$ and ...
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30 views

How do I show the tangent to an elliptic curve over the complex numbers meets the elliptic curve at another point?

If $E(\mathbb{C})$ is an elliptic curve given by $y^2=ax^3+bx+c$ for $a,b,c\in \mathbb{C}$, and $\ell$ is a line tangent to $E(\mathbb{C})$ at some point $p$, then why does $\ell$ meet $E(\mathbb{C})$ ...
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1answer
43 views

Is the dual of the tautological subbundle on a grassmannian ample?

Consider the Grassmannian $G(k,V_{n})$ of k-dimensional subspaces in an n-dimensional vector space $V_{n}$. We have the "restriction" map of vector bundles $V_n^* \rightarrow \mathcal{E}_k$, where $...
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0answers
15 views

In a DVR, why does $u=f(t,u)$, with $f$ a homogeneous polynomial of degree $3$ and $t$ a uniformizer, imply $\nu(u)=3$?

In this answer by Georges Elencwajg, it is stated that $$u=t^3-\dots-e_1e_2e_3u^3=\text{a homogeneous polynomial of degree $3$ in t,u}\quad(\ast)$$ [...] Now in the local ring $\mathcal O_{...
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18 views

Order of point in divisor

I would like to prove that for hyperelliptic curve $y^2=f(x)$ over $k$ and for $a,b\in k$ divisor of the function $g(x,y)=\frac{x-b}{x-a}$ is equal to $$div(g)=(b,\sqrt{f(b)})+(b,-\sqrt{f(b)})-(a,\...
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32 views

$\operatorname{Pic}^0$ of a product of two varieties

Let $X,Y$ be two algebraic varieties over a field $K$. I define $\operatorname{Pic}^0(X)$ as a group of divisors algebraically equivalent to zero. Is it true that $\operatorname{Pic}^0(X \times Y) = \...
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0answers
29 views

How to define the Zariski tangent space of a divisor?

I'm interested in the case where $X$ is an abelian variety, so we have the correspondence between line bundle and divisors (and I like to think in terms of Weil divisors). Let's first recall the ...
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1answer
30 views

Calculating a composition of finite correspondences

This is part of exercise 1.11 in Mazza, Voevodsky, Weibel's Lecture Notes on Motivic Cohomology. Here's some notational stuff: we're working in the category $Cor_k$ whose objects are the smooth ...
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23 views

Derived pushforward of exceptional divisors

Let $f: X \to Y$ be a birational morphism between projective, smooth varieties. Suppose $E$ is an effective exceptional divisor of $f$. Then it is well known that $f_*(\mathcal{O}_X(E)) = \mathcal{O}...
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1answer
84 views

Twisted sheaf isomorphic to invertible sheaf associated to 1-cocycle

Let $X=\operatorname{Proj}A$ for some graded ring $A$ such that the irrelevant ideal $A_+$ is generated by the degree one part $A_1$ of $A$. Then Bosch's $\textit{Algebraic Geometry and Commutative ...
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1answer
71 views

Understanding a theorem in algebraic geometry

Theorem: Let $X$ be projective, $Y$ be quasi-projective variety, and $f: X \to Y$ be an algebraic morphism. Then the image $f(X)\subset Y$ is Zariski closed. I am trying to understand my note that ...
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74 views

Algebraic Geometry: Flatness is equivalent to being scheme-theoretically dense?

In Mumford's penultimate draft for his Algebraic Geometry book, he says in p.144 (followed by a definition 4.12 which I copied here): Def 4.12. Let R be a valuation ring, and let $\eta$ (resp. $o$)...
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15 views

number of irreducible components [duplicate]

In some cases, we can draw a variety and just count the number. For example, $xy =0$ in $k[x,y]$. Also, there are many results in algebraic geotmetry or algberaic groups, telling one how to prove the ...
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47 views

Why is this intersection supported on the closed point?

Let $R$ be a (commutative unitary) local ring. Let $M$ and $N$ be finitely generated $R$-modules such that $\mathrm{length}(M\otimes_R N)$ is finite. Let $x$ be the closed point of $X=\operatorname{...
2
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1answer
60 views

the defining polynomials of $PGL_n$ as an affine algebraic group

I have read this question. Also, there are theorems telling me that $PGL_n$, as the quotient of $GL_n$ by its center, is with no doubt an affine variety (affine algebraic group). But, is it true that ...
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0answers
27 views

$Q$ parabolic in $P$, $P$ parabolic in $G$ implies $Q$ parabolic in $G$.

I am a bit confused on the proof of this lemma. $G$ is a linear algebraic group over an algebraically closed field $k$. A closed subgroup $P$ of $G$ is called parabolic if the quotient variety $G/P$ ...
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0answers
33 views

Effective divisor of degree $2$ over finite field and number of points in Jacobian

I am given a hyperelliptic genus $2$ curve $C\dots y^2=f(x)$ over the finite field $\mathbb{F}_q$ and I need to prove that $$\#J(\mathbb{F}_q)=\frac{1}{2}\#C(\mathbb{F}_q)^2+\frac{1}{2}\#C(\mathbb{F}...
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1answer
73 views

Morphisms of sheaves v.s. morphisms of presheaves

Let $\mathcal{F},\mathcal{G}$ be presheaves on a topological space $X$ and $\alpha$ a morphism of presheaves. Consider the following four statements. I meant here "for all $U$ open in $X$" 1)$\alpha$ ...
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2answers
42 views

References for the functor of points vision of schemes.

I recently discovered the idea of the functor of points. I would like to find a reference where the different visions of scheme are presented. It seems to me that the classical texts emphasize the ...
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0answers
35 views

Connected component of $0$: why is it an abelian variety?

With "Abelian variety" I mean a integral scheme $X$, proper over an algebraically closed field (complete variety) with a group law $m: X\times X \rightarrow X$ such that $m$ and the inverse map are ...
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2answers
68 views

What happens with the empty set in the categorical definition of a presheaf?

I've found in Hartshorne's book on Algebraic Geometry a categorical definition of a presheaf (Ch. 2.1, just after the first definition). He there defines for a topological space $X$ the category $\...
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1answer
47 views

Center and axis of Quadratic Surface

Given a Quadratic Surface in the form: $ax^2+by^2+cz^2+2fyz+2gzx+2hxy+2px+2qy+2rz+d=0$ I know how to decide which kind of surface is represented (http://mathworld.wolfram.com/QuadraticSurface.html). ...
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1answer
43 views

Line bundle and effective divisor?

$X$ is for me an abelian variety. I think I have seen this result somewhere but I can not find it anymore: For any line bundle $\mathcal{L}$, $\dim H^0(X,\mathcal{L})>0$ if an only if $\...
2
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2answers
49 views

Locally free sheaf of rank n on $A^1_k$ is trivial of rank n

The question is how to prove the title, this is actually exercise 13.2.C. from Vakil's notes. The hint is to use the structure theorem for f.g. module over PID. Since to be quasi-coherent is a local ...
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1answer
30 views

Understanding $X \to X^{tt}$ from the Poincare bundle

Let $X$ be an abelian variety over $k$, $X^t = \text{Pic} _{X/k}^0$ its dual, and $\mathscr{P}$ be the Poincare bundle on $X \times X^t$. View $\mathscr{P}$ as a family of line bundles on $X^t$ ...
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31 views

Closed point in the generic fiber of an arithmetic surface

Let $S$ an irreducible Dedekind scheme of dimension $1$, and let $\pi:X\to S$ be a regular, integral fibered surface. We assume that $\pi$ is a flat morphism and that $X_\xi$ is the generic fiber over ...
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37 views

Image of finite locally free morphism is open

Let $f$ be a finite and locally free morphism $\phi :X \longrightarrow S$. In Szamuely (Galois groups and fundamental groups), it is said that the image of $\phi$ is open. The proof of Szamuely goes ...
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29 views

Curves on $\mathbb{P}^1\times\mathbb{P}^1$

For curves in a projective plane there is a degree-genus formula. Is there an analogue for curves on $\mathbb{P}^1\times\mathbb{P}^1$? More specially, is there a criterion (in terms of genus and ...