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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Why is every open in $\mathbb{A}^1$ necessarily principal?

Let $U\subseteq\mathbb{A}^1$ be an open set in affine $1$-space. Why is $U$ necessarily a principal open set? Since $U$ is the complement of a closed set, I write $U=\mathbb{A}^1\setminus V(S)$ for ...
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Transformations from n-sphere coordinates to cartesian coordinates.

I was wondering how one would proceed to convert between coordinate systems in $ \mathbb R^n $. For $ \mathbb R^2 $ the conversion is easy and just basic trigonometry. Given $(r, \theta)$ we can ...
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17 views

Computing these multiplicities

I'm trying to use some Algebraic Geometry techniques to check my understanding on them. I'm using the most stupid of all the examples: trying to compute the multiplicities of the intersections of the ...
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Converse of the implication $V(S)\subseteq V(T)\iff T\subseteq\sqrt{\langle S\rangle}$.

I'm having trouble recalling one direction of the following bi-implication. Suppose $S,T$ are subsets of the polynomial ring $k[X_1,\dots,X_n]$ over an algebraically closed field. We have ...
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17 views

Rational points of $ax^2+by^2=z^r$, $r $ odd integer.

I am trying to find the rational points of:$$ax^2+by^2=z^r$$ I am aware that:$$(u^r-2^{r-2}v^r)^2+(2uv)^r=(u^r+2^{r-2}v^r)^2$$ How can I deduct the results?
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Geometric meaning of intersection multiplicities?

I am wondering about the geometric significance of the intersection multiplicity of two curves as defined in Hartshorne 5.4 (The length of $O_p/(f,g)$ is the intersection multiplicity of $Z(f)$ and ...
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21 views

Elimination theory in Hartshorne

Does anyone know a good reference for elimination theory (Theorem 5.7A) mentioned in Hartshorne? The reference he gives is Van der Waerden modern algebra volume two, but it didn't feel locally ...
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17 views

Mild singularities in Algebraic variety

I am looking for the mathematical definition of mild singularities. Is there any refference for following : Varieties obtained from divisorial contraction and flips have mild singularities.
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24 views

Cohomology of pullback of a vector bundle along a closed embedding

Let $f : Z \hookrightarrow X$ a closed embeding of smooth projective varieties, and let $V$ be a vector bundle over $X$. Are there general facts that allow one compute cohomology and chern classes of ...
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20 views

about the spanned divisor of a complex variety

I have this definition: let $\xi \in H^1(X,O^*)$ a cocycle. We say that $\xi$ is spanned if for every point $x$ in my variety $X$ there exist a section $s \in H^0(X,O(\xi))$ such that $s(x) \neq 0$. ...
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are base change and restriction of scalars “inverses” in this case?

Let $l/k$ be a finite extension of fields. Let $G$ be an affine $k$-groups scheme. Let $G_l = G \times_k l$. Is it true that $\mathfrak{R}_{l/k}(G_l) = G$, where $\mathfrak{R}_{l/k}(-)$ is the ...
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24 views

When the fibers of a flat morphism are varieties.

Notations: As in Hatshorne's book. Suppose that $f:X\longrightarrow Y$ is a flat morphism between two non-singular projective varieties over an algebraically closed field. Are the fibers of $f$ ...
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25 views

Perimeter of an ellipse intuition help

I am aware that you can take the circumference of an ellipse using an elliptic integral and haven't looked much into it, but that seems to be a bit extreme and i was taking a personal look at conic ...
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19 views

$X\to \textrm{End}(O_{X,e}/m_{X,e}^r)$ is a morphism

Suppose $X^n$ is a complete group variety over algebaically closed field $k$, then the group law can be shown to be commutative. In proving this, one step is to show $X\to ...
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16 views

Intersection of open affines in a prevariety

What is an example of a prevariety in which the intersection of some two open affines is not an open affine? My examples of prevarieties that are not varieties does not extend beyond the affine line ...
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75 views

Etymology of “flabby” or “flasque” sheaf

I just started working with flasque, or flabby sheaves, that is sheaves whose restriction maps are surjective for any two open set of the space. I wonder about the etymology of the term. In French, ...
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47 views

Why aren't those Cartier Divisors equivalent?

Please refer to Gathmann's notes http://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/main.pdf at Example 9.3.6 for context. It's trying to give an example that the map between $Div(X)$ and ...
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43 views

Intersection of Segre variety with linear spaces

Consider the intersection of the Segre variety associated to product of $n$ copies of $\mathbb P^2$, with $k$ linearly independent hyperplanes. Is it possible to drop one of the hyperplanes and obtain ...
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1answer
21 views

Iitaka fibration over canonical model

I am looking for a referrence for the proof of following fact If the minimal projective manifold has positive Kodaira dimension and it is not of general type, it admits an Iitaka fibration over ...
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36 views

A particular example of a non-reduced scheme (with a reduced ring of global sections)?

I am searching for a scheme $X$ which can be obtained by gluing two affine schemes (along open subsets) such that: 1) X is non-reduced; 2) $\Gamma(X,\mathcal O_X)$ is a reduced ring. Any ideas? ...
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34 views

Quick question: Chern classes of Sym, Wedge, Hom, and Tensor

Given $L$ is a line bundle and $V$ is bundle of rank $r$ on a surface (compact complex manifold of dim 2). Recall the formula for $c_1$ and $c_2$: $c_1(V\otimes L)=c_1(V)+rc_1(L)$ ...
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About the holomorphic vector field tangent to a divisor

We say a holomorphic vector field $X$ is tangent to an effective divisor $D$, if $D_Xs=\lambda s$, where $s$ is the determining holomorphic section of the line bundle $L_D$ corresponding to $D$. If ...
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22 views

find points on circle in 3D pace perpendicular to line

I'm working with 3D image data and have little algebraic knowledge. I have an 3D image whit each pixel divined by its x,y,z position. What I need is to get the values of all pixels on a circle inside ...
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21 views

The vector bundle of an hypersurface

Suppose that $X$ is a compact complex variety and $V \subset X$ an irreducible hypersurface. Let $\{U_{\alpha}\}_{\alpha \in I}$ an open covering of X. With $s_{\alpha}$ i denote the local equation of ...
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Why is the function defined by $f(x_1,x_2)=0$ when $x_1=0$, and $x_2$ when $x_1\neq 0$ not regular?

I'm having trouble understanding what should be a straight forward example. Suppose $X\subseteq\mathbb{A}^2_k$ is cut out by the equation $x_1(x_2^2-x_1)=0$. Define a function $f:X\to k$ (here ...
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What happens when you drop “étale” from the construction of étale fundamental groups

Haven't thought about this deeply; it just came up at lunch and I wondered if anyone knew the answer. To define the étale fundamental group of a scheme $X$, we fix a point $x_0 \in X$, look at the ...
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36 views

Fibres of an ideal sheaf , total spaces and torsion groups

My question concerns a common example, which seems to often appear as an example/counter-example. Let $k$ be a field and consider the ideal exact sequence of the structure sheaf $k(p)$ of a point $p$ ...
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34 views

Zariski-open subset in $\mathbb{C}^n$ to Zariski-closed subset in $\mathbb{C}^{n+1}$

Let $n \in \mathbb{N}$, $f\in \mathbb{C}[X_1,\dots,X_n]$, and $D(f):=\{x=(x_1,\dots,x_n)\in \mathbb{C^n}|f(x)\neq 0\}.$ I want to show that there is a injective $\Phi: D(f) \rightarrow ...
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24 views

What is the $\pi_1$-action on the hom-sheaf between two finite etale covers?

Say you have two finite etale covers $X\rightarrow S$, $Y\rightarrow S$. The hom sheaf $\mathcal{H}om_S(X,Y)$ on the etale site $\text{Sch}/S$ is finite locally constant, hence representable by some ...
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Genus of intersection of two surfaces in $\mathbb{P}^3$

Let $F_1$ and $F_2$ be two (smooth) surfaces in $\mathbb{P}^3$, of degrees $d_1$ and $d_2$ respectively. Let $C$ denote curve given as their intersection. How one can compute arithmetical genus of the ...
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50 views

Is Spec ($k[x_1,x_2,\ldots])$ a smooth $k$-scheme?

Let $k$ be a field and let $A = k[x_1,x_2,\ldots]$. Note that $A$ is not of finite type over $k$. Is $\operatorname{Spec} A\to \operatorname{Spec} k$ a smooth morphism of schemes? I think it is ...
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1answer
53 views

Categories and the direct image functor

I've just started learning about sheaves and yesterday I had a look at MacLane/Moerdijk's book "Sheaves in Geometry and Logic". I've discovered something in this book on which I'd like some ...
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The interplay between projective and affine varieties.

I'm studying Algebraic Geometry first course from Harris and I didn't understand this equality: In another words, I'm having troubles to understand the interplay between $f_{\alpha}$ and ...
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1answer
45 views

generalized Euler exact sequence

I'm reading about Euler exact sequence in Ravi Vakil's notes, and I need help to check a few things. Given a scheme $X$, and a locally free sheaf $\mathcal{E}$ of rank $n+1$ on $X$, let us start from ...
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1answer
28 views

Fulton 8.17 ¿$\Gamma(X) = k$?

Problem 8.17 Let $X, Y$ be nonsingular projectives curves, $ f:X\to Y $ a dominating morphism. Prove that $ f (X)=Y $. Im trying to solve the problem 8.17 of Algebraic Curves Book of Fulton, there ...
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22 views

sheafification construction in Hartshorne

In section II.1 of Hartshorne, the sheaf $\mathscr F^+$ associated to a presheaf $\mathscr F$ is constructed so that $\mathscr F^+(U)$ is the set of functions $$ s\colon U \to \bigcup_{p \in U} ...
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43 views

The blow-up of $ X $

I'm studying blow-ups in connection with an introduction course in algebraic geometry. I've some problems with the details in the below set-up, which my textbook introduces in order to define the blow ...
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19 views

If the saturated ideal of hypersurface generated by one element?

If a subscheme of codimenion one in $\mathbb{P}^n_k$ is define by ideal sheaf $\mathcal{I}$, is the saturation $\oplus_{n\ge 0}\Gamma(I(n))\subseteq k[x_0,...,x_n]$ be generated by one element? Is ...
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35 views

Is the sum of saturated ideals saturated?

In a graded ring $S=\oplus_{k=0}^{\infty}S_k$, denote $m=\oplus_{k=1}^{\infty}S_k$, call an ideal $I$ to be saturated if $I=\cup_{n=1}^{\infty}(I\colon m^n)$. Is the sum of two saturated ideals still ...
2
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1answer
41 views

Rational functions on varieties

To give a rational function $f$ in the function field $K(X)$ of a smooth projective curve $X$ is equivalent to giving a finite morphism $f:X\to \mathbb P^1$. What is the precise analogue of this ...
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26 views

Rational functions over variety X

I 'm trying to solve this exercise of Fulton Algebraic Curves: Let $D=\sum n_P P$ be an effective divisor, $S=\{P \in X: n_P>0\}$, $U=X\setminus S$. Show that $L(rD)\subset ...
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What is $\mathrm{Proj}(S \otimes R)$?

What the title says. Let $S$ be a graded $k$-algebra, generated in degree $1$, and the same for $R$. Then $S \otimes_k R$ is graded as well, with $k$'th graded piece $\bigoplus_{j+l=k} S_j \otimes_k ...
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Definition of intersection multiplicity of a curve with some hyperplanes

I'm studying the chapter 2 of this paper and I have the following doubt: What is the definition of intersection multiplicity of a curve $C$ with some hyperplanes at a point $P$? Remark: My only ...
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28 views

Something like the Weierstrass preparation theorem?

I'm reading Griffiths and Harris, On the Noether-Lefschetz Theorem and Some Remarks on Codimension-two Cycles, Math. Ann. 271, 31-51 (1985). Their proof of the Noether-Lefschetz theorem is based on ...
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1answer
36 views

Confused about short exact sequence involving $\mathcal{O}_{\mathbb{P}^n}$ and $\mathcal{O}_Z$ for $\pi:Z\rightarrow \mathbb{P}^n$ closed embedding

If $\pi: Z\rightarrow \mathbb{P}^n$ is a closed embedding where $Z$ is the zero set of some degree homogeneous polynomial, we have: $$0\rightarrow \mathcal{O}_{\mathbb{P}^n}(-d)\rightarrow ...
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65 views

Why is such a the series algebraic but rational?

The coefficients of the series expansion of the algebraic function $A=\frac{1-\sqrt{1-8x^2}}{4x}$ are all intergers: $$A(x)=x+2x^3+8x^5+\cdots$$ But according to Polya's research,if $ F(x)$ is a ...
2
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1answer
39 views

First Chern class of line bundle corresponding to divisor

If I know an effective divisor $D$, then there is a line bundle $L_D$ corresponding to this divisor. How can I compute the first Chern class of $L_D$? For example, on $\mathbb{C}\mathbb{P}^3$, ...
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2answers
58 views

Stalks of the sheaf $\mathscr{H}om$?

The question is basically the title. What are the stalks of the sheaf $\mathscr{H}om_{\mathscr{O}_X}(\mathscr{F},\mathscr{G})$? If $X$ is a noetherian scheme and $\mathscr{F}$ is coherent, then ...
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1answer
37 views

Does a projective variety have a torus fixed point?

Let $X$ be a projective variety over $\mathbb{C}$ and let $T=(\mathbb{C^*})^k$ act on it. Is it true that there is a fixed point of this action on every irreducible component of $X$ just because $X$ ...
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40 views
+50

Pullback of principal Cartier divisors along a field extension

I tried the following problem in Liu's book, 7.3.1 but I don't see where it was needed that $X$ is integral - maybe someone can help me here. Is the following true without supposing that $X$ is ...