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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143 views

Hom functor of quasi-coherent sheaf maybe not quasi-coherent

I notice that some books say that for arbitrary quasi-coherent sheaves $F$, $G$ over a scheme $X$, the $\mathcal O_X$-module $\mathrm{Hom}_{\mathcal O_X}(F,G)$ maybe not quasi-coherent, who can give ...
4
votes
1answer
43 views

Derivations are determined by their values on linear functions

How are derivations of the $\mathbb R$ algebra of germs of differentiable real functions on a manifold completely determined by their values in germs of linear functions? Are derivations of more ...
4
votes
1answer
254 views

Theorem 5.1. Chapter I in Hartshorne Book's

I find difficulty to understand the proof of this theorem : Theorem : Let $Y\subseteq\mathbb A^n$ be an affine variety. Let $Ρ\in Y$ be a point. Then $Y$ is nonsingular at $Ρ$ if and only if the ...
4
votes
2answers
133 views

$Z(I:J)$ is the Zariski closure of $Z(I)-Z(J)$

Let $(I:J)$ denote the colon ideal (or ideal quotient). It is pretty clear that the Zariski closure of $Z(I)-Z(J)$ is contained in $Z(I:J)$. How can we prove that the the Zariski closure of $Z(I)-Z(J)$...
4
votes
1answer
85 views

Question of the relation between very ampleness and irreducibility

Let $X$ be a projective surface and $D$ be a divsor. Then I know $D$ correspond to a curve of $X$. My qeustion is simple. If $D$ is very ample, then the corrsponding curve of $D$ is irreducible? More ...
4
votes
2answers
151 views

Hartshorne II Prop. 6.9

Prop. 6.9: Let $X \to Y$ be a finite morphism of non-singular curves, then for any divisor $D$ on $Y$ we have $\deg f^*D=\deg f\deg D$. I can not understand two points in the proof: (1) (Line 9) ...
4
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1answer
76 views

Ampleness, Nakai's criterion and pullback

In the book I'm reading ( Geometry of Algebraic Curves ), at some point (page $310$) they make the following claim: One can use Nakai's criterion to establish the general fact that if $f:X\to Y$ ...
4
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1answer
73 views

Examples that the morphism $X\times_k k' \rightarrow X$ is not closed

Let $k$ be a field. Let $k'$ be an extension field of $k$. Let $X$ be a $k$-scheme of finite type. If $k'$ is algebraic over $k$, the morphism $X\times_k k' \rightarrow X$ is integral. Hence it is ...
4
votes
2answers
362 views

General quadratic diophantine equation.

Here is my problem: I am given a general quadratic diophantine equation: $$ax^2 + bxy + cy^2 + dx + ey + f = 0$$ where $x$ and $y$ are variables with integers $a,b,c,d,e,f$. I have to show that if the ...
4
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1answer
440 views

Why does the degree of a line bundle equal the degree of the induced map times the degree of the image plus the degree of the base locus?

Let $L$ be a line bundle on a smooth curve $C$. If $L$ is rank $r+1$, define the induced map (as Arbarello, Cornalba, Griffiths, Harris): $$\begin{aligned}\phi :& C \rightarrow \mathbb P|L|^*\\&...
4
votes
1answer
115 views

Finding the Vanishing Set of an Algebraic Set

We've been given the set $X = \{(t^3,t^4,t^5) \in \mathbb{A}^3 \mid t \in \mathbb{A}^1\}$ (where the underlying field $\mathbb{K}$ is infinite), and have been asked to show that $X = \mathbb{V}(J)$ ...
4
votes
1answer
253 views

Motivation and Applications for Toric Varieties

I'm a graduate student of mathematics starting to study algebraic geometry with a focus on toric varieties (along Cox, Little, Schenk). From what I learned so far, I can grasp that toric varieties ...
4
votes
1answer
245 views

Limits and colimits in the category of schemes

What is the smallest category enlarging the category of schemes over a field $k$ which is: Complete? Cocomplete? Admits a cogenerator? generator? I admit there is some overlap with my previous ...
4
votes
1answer
154 views

Identifying a line bundle on $\mathbb{P}^1$

I have a geometric line bundle $L$ on $\mathbb{P}^1 = \{[x_0:x_1]\}$. With respect to the standard affine cover $U_0 = \{x_0 \neq 0\}$ and $U_1 = \{x_1 \neq 0\}$, I have the transition function $[x_0:...
4
votes
1answer
172 views

Compact variety which is not projective

While reading Andreas Gathmann's notes on Algebraic Geometry, I stumbled upon this statement: "Projective varieties form a large class of “compact” varieties that do admit such a unified global ...
4
votes
1answer
92 views

Isomorphisms of $\mathbb P^1$

Prove that every isomorphism of $\mathbb P^1$ (over an algebrically closed field $\mathbb K$) is of the form $$ \phi(x_0: x_1) = (ax_0+bx_1 : cx_0 + dx_1) $$ where $\begin{pmatrix} a & b \\c &...
4
votes
1answer
654 views

Cremona transformations are birational maps

Consider the following map, which is a Cremona transformation: $$ \begin{split} f\colon & \mathbb P^2 \dashrightarrow \mathbb P^2 \\ & (x:y:z) \mapsto (xy: xz: yz) \end{split} $$ I have to ...
4
votes
1answer
140 views

Question regarding Hartshorne Example II.(6.5.2)

Let $k$ be a field, let $A=k[x,y,z]/\langle xy-z^2\rangle$ and let $X=\operatorname{Spec}A$. Let $Y:y=z=0$ I want to know the divisor of $y$ In Hartshorne book, because $y=0 \Rightarrow z^2=0$ and $z$...
4
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1answer
169 views

Hartshorne III 9.8 understanding associated points and extensions

From Hartshorne III Prop 9.8 Let Y be a regular scheme of dimension 1, let P $\in Y$ be a closed point, and let X $\subset \mathbf P_{Y-P}^n$ be a closed subscheme which is flat over Y - P. Then ...
4
votes
1answer
160 views

Genus over finite fields

Is there a way of computing the genus of a parametrized curve over a finite field? For instance I am interested in the genus of the following space curve in the m-dimensional space over $F_{q^k}$ ...
4
votes
1answer
175 views

Intersections of tangents with cubic are colinear

I am trying to do Exercise 5.33 on page 64 of Fulton's book on algebraic curves. 5.33 Let $C$ be an irreducible cubic, $L$ a line such that $L\bullet C = P_1+ P_2 + P_3,$ $P_i$ distinct. Let $L_i$ ...
4
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1answer
218 views

Morphisms between locally ringed spaces and affine schemes

I need some hints to understand the conclusion of the proof of the following lemma from the Stacks Project: Lemma $\mathbf{6.1.}\,$ Let $X$ be a locally ringed space. Let $Y$ be an affine scheme. ...
4
votes
1answer
273 views

Classifying map

Let $\xi=(E,p,B)$ a principal $G$-bundle and $\eta=(P,\pi,Q)$ a real vector bundle such that $\operatorname{rank}(\eta)=n$. We can consider a classificant space $BG$. What is the classifying map $f:X \...
4
votes
1answer
59 views

Quasicompact over affine scheme

Let $X$ be a scheme and $f : X \rightarrow \mathrm{Spec}\, A$ a quasicompact morphism. Are there any easy conditions on $A$ under which we can say that $X$ is quasicompact? Quasicompact morphism ...
4
votes
1answer
639 views

When is the pullback functor on sheaves faithful?

For a flat finite surjective morphism of smooth varieties $f : X \rightarrow Y$ we have the pushforward functor $f_* : \mathcal{S}h (X) \rightarrow \mathcal{S}h (Y)$ and its left adjoint $f^* : \...
4
votes
1answer
188 views

What is the definition of “local equation(s)” for a subscheme?

Hartshorne mentions "local equations" a few times without (so far as I can tell) actually defining them anywhere. As best as I can guess, the definition would be something like this: If $Y \...
4
votes
1answer
82 views

studying the topology of a real algebraic set

Let $f_1,\ldots,f_n \in \mathbb{R}[x_1,\ldots,x_m]$ be polynomials with real coefficients and let $I$ be the ideal that they generate. Denote by $V_{\mathbb{R}}(I)$ the corresponding real variety, i.e....
4
votes
1answer
147 views

Segre varieties contained in hyperplanes

Recall, the Segre embedding is a map $\sigma: \mathbb{P}^m \times \mathbb{P}^n \to \mathbb{P}^{(m+1)(n+1)-1}$ given by \begin{equation} ([x_0:\cdots:x_m],[y_0:\cdots:y_n]) \mapsto [x_0y_0:x_0y_1: \...
4
votes
1answer
373 views

Pullback of sheaves and pullback of schemes

Let $\mathbb{G}_m$ the multiplicative group, with coordinate ring $\mathbb{C}[x^{\pm 1}]$, and considered as a sheaf of abelian groups over $\mathrm{Spec}\,\mathbb{C}$ in the Zariski topology. Let $X$ ...
4
votes
1answer
146 views

Toric Varieties from Cones

Consider the lattice $N=\Bbb{Z}^d$ spanned by $e_1,\dots,e_d$ and the cone $$\sigma=\text{Cone}\{e_1,\dots,e_k\}, \quad k<d.$$ I am trying to understand why the toric variety $V_\sigma$ obtained is ...
4
votes
1answer
302 views

The image of the diagonal map in scheme

Let $X\rightarrow S$ be a separated morphism of schemes, that is, the diagonal map $\Delta:X\rightarrow X\times_S X$ is a closed inmersion. In general (see Closure of image of diagonal morphism of S-...
4
votes
3answers
157 views

When are the sections of the structure sheaf just morphisms to affine space?

Let $X$ be a scheme over a field $K$ and $f\in\mathscr O_X(U)$ for some (say, affine) open $U\subseteq X$. For a $K$-rational point $P$, I can denote by $f(P)$ the image of $f$ under the map $$\...
4
votes
1answer
390 views

Koszul complex of locally free sheaves

Let $X$ be a complex variety; one can also assume it is smooth if this helps. $\mathcal{E}$ is a locally free sheaf of rank $r$ on $X$, and $s \in H^0(X, \mathcal{E})$. Then one has a Koszul complex ...
4
votes
2answers
116 views

How can we check the gluing property of sheaf of ideals?

For a ringed space $(X,\mathcal{O}_X)$, one can define a sheaf of ideals $\mathcal{J}$ of $\mathcal{O}_X$. Then how can we see the $\mathcal{J}$ satisfies the conditions of sheaf? Especially, I cannot ...
4
votes
1answer
96 views

Direct image for standart affine cover of projective line

Let $U_1 \cong Spec(K[t])$ and $U_2\cong Spec(K[t])$ be a standard affine cover of a projective line $\mathbb{P^1}(K)$, where $K$ is some field. Let us denote open embedding as $j_k$ $$ j_k : U_k \to \...
4
votes
1answer
140 views

Property kept under base change and composition is preserved by products

The following is true? Why? Let $P$ be a property of morphisms preserved under base change and composition. Let $X\to Y$ and $X'\to Y'$ be morphisms of $S$-schemes with property $P$. Then the unique ...
4
votes
1answer
125 views

Irreducibility preserved under étale maps?

I remember hearing about this statement once, but cannot remember where or when. If it is true i could make good use of it. Let $\pi: X \rightarrow Y$ be an étale map of (irreducible) algebraic ...
4
votes
1answer
765 views

Normalisation of an algebraic curve.

I need to compute explicitly the normalisation of a singular algebraic curve $C$ which is given by an explicit equation in $\mathbb{A}^2$. This task is mostly reduced to finding the integral closure $\...
4
votes
1answer
662 views

Calculating the distance between a camera and a target using camera output

I have a 640x480 camera that recognizes a rectangle that is 1ftx2ft. Is it possible to calculate the distance between the camera and the rectangle? Edit: The horizontal angle of view is 54°.
4
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1answer
166 views

Basic Question on the Ideal-Variety Correspondence and Adjoint Functors

Let $k$ be an algebraically closed field. The ideal-variety correspondence says that the equations \begin{align} \mathbf{I}(X) &= \left\{f\in k[x_1,\dotsc,x_n]:p\in X\Rightarrow f(p)=0\right\} \\ \...
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votes
1answer
193 views

Is a surjective homomorphism of regular local rings necessarily an isomorphism?

Let $R$ and $S$ be regular local rings, and $f: R\rightarrow S$ a surjection that induces an isomorphism on tangent spaces. Is $f$ necessarily an isomorphism? I believe the answer should be yes, ...
4
votes
2answers
427 views

Tangent spaces of affine algebraic varieties at singular points

Let $X$ be an affine algebraic over the algebraically closed field $k$ and let $\mathcal{O}(X)$ be the ring of its regular functions. Let us assume that $X$ is irreducible and let $x\in X$. There ...
4
votes
1answer
572 views

Why surjectivity stable under base change?

I want to prove that surjectivity is stable under base change: if $f:X\to S$ a surjective morphism of scheme and $\varphi:T\to S$ then $f_T:X\times_S T\to T$ is surjective. Idea 1: I know that for ...
4
votes
1answer
205 views

Is a morphism between schemes of finite type over a field closed if it induces a closed map between varieties?

This is the converse of this question. Let $X$(resp. $Y$) be a scheme of finite type over a field $k$. Let $f\colon X \rightarrow Y$ be a morphism. Let $X_0$(resp. $Y_0$) be the set of closed points ...
4
votes
1answer
120 views

Irreducible trivialization of a finite etale morphism

Let $X$ be an irreducible scheme and $Y \to X$ a finite étale morphism. Is there some finite étale cover $Z \to X$ which trivializes $Y$ (i.e. $Y \times_X Z$ is a union of copies of $Z$) such that $Z$ ...
4
votes
1answer
150 views

Cartier divisor and Dimension of Cohomology Group

I am doing some practice questions for my exam and I would appreciate help in solving this problem: $D,E$ are Cartier divisors on a nonsingular projective surface $X$. (1) If $D\equiv 0$ show ...
4
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1answer
55 views

Very ampleness of $\omega_{C}^n$

Let $C$ be a genus $g$ curve over complex numbers. How can I prove that $\omega_{C}^n$ is very ample for $n\ge2$ if $g=2$ and $n\ge 3$ if $g\ge 3$? Also, I wonder if this still true for other fields ...
4
votes
1answer
80 views

$\mathcal{O}_{X}(d)\simeq \mathcal{O}_{X}(D)$?

On $\mathbb{P}^n$ let $D$ be a smooth hypersurface defined by the equation $F=0$, F an homogeneous polynomial. $\mathcal{O}_{\mathbb{P}^n}(D)$ is the sheaf of meromorphic functions on $\mathbb{P}^n$ ...
4
votes
1answer
168 views

Is $Z(x^2-y^3)$ isomorphic to $Z(y^2-x^3-x^2)$ over the complex numbers?

I'm having trouble determining if the algebraic sets $Z(x^2-y^3)\subset \mathbb{A}^2$ and $Z(y^2-x^3-x^2)\subset\mathbb{A}^2$ are isomorphic over $\mathbb{C}$. My guess is that this boils down to ...
4
votes
1answer
438 views

When the the presheaf of image of morphism of sheaves is a sheaf?

For given a morphism of sheaves, in general, I know that the presheaf of image(or the presheaf of cokernel) is not a sheaf. is there when the the presheaf of image(or the presheaf of cokernel) of ...