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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2
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2answers
94 views

Why the unitary group is not a complex algebraic variety?

The question comes from Exercise 1.1.2 of the book "An Invitation to Algebraic Geometry". By definition the unitary group U(n) is the group of all complex matrix that satisfies $U^*U=I$. I know that ...
1
vote
0answers
9 views

Local description of Frobenius trace map

I'm trying to understand the local description for the Frobenius trace map. Let $X$ be a smooth projective variety. The Frobenius map acts on the topological space of $X$ as the identity and acts on ...
4
votes
2answers
68 views

Rational Points, classical versus modern notion

In classical algebraic geometry, a $\mathbb Q$-rational point on a, say, complex affine variety $V\subseteq\mathbb C^n$ is a point $p=(p_1,\ldots,p_n)$ with $\forall i: p_i\in\mathbb Q$. Now, in ...
1
vote
1answer
81 views
+50

A possible mistake in Hartshorne chapter 2 proposition 2.6

Here is the context of this question. Hartshorne claim that $O_X(U)\cong \beta_*(O_V)(U)=O_V(\beta^{-1}(U))$ for any open $U\subset X=\operatorname{Spec}A$,but it is possible that ...
1
vote
1answer
36 views

maple plot of Belyi function

I would like to understand how to construct Figure 5 of the paper Composition is a generalized symmetry by Alexander Zvonkin: The hypermap/dessin d'enfant of Figure 4 is while the Belyi function ...
1
vote
0answers
10 views

isomorphism in étale cohomology

Let $X/k$, $k$ algebraically closed, be a proper variety. Why is $H^2_{et}(X,\mathbf{Z}_\ell) \otimes \mathbf{Q}_\ell/\mathbf{Z}_\ell = (\mathbf{Q}_\ell/\mathbf{Z}_\ell)^{b_2(X) - \rho(X)}$, where ...
1
vote
0answers
11 views

finiteness in étale cohomology (ell-adic sheaves)

Let $X/k$, $k$ separably closed, be a proper variety. Is then $H_{et}^p(X,\mathbf{Z}_\ell(q))$ finitely generated?
0
votes
0answers
21 views

Is there a 4 pointed star that is regular?

I am studying about the area of a 4 pointed star, I wonder if there is really a 4 pointed star that is regular? what could be the characteristics of a regular 4 pointed star?
14
votes
3answers
591 views
+100

When do equations represent the same curve?

Suppose we have two sets of parametric equations $\mathbf c_1(u) = (x_1(u), y_1(u))$ and $\mathbf c_2(v) = (x_2(v), y_2(v))$ representing two 2D planar curves. When I say "2D planar curves" I mean ...
0
votes
0answers
25 views

Hyperplanes as dual projective spaces

I was reading through Harris's Algebraic Geometry book, and was slightly perplexed by the following paragraph: "Note that the set of hyperplanes in a projective space $\mathbb{P}^{n}$ is again a ...
0
votes
1answer
23 views

Is projection $\mathbb{A}^2 \to \mathbb{A}$ finite?

I am misunderstanding something very basic here. We know a regular map $\varphi : W \to V$ is finite iff $k[W]$ is a finite $k[V]$-algebra. We also know that finite morphisms have finite fibers. ...
0
votes
0answers
16 views

Calculating ramification points

I was looking at the following example/explanation Understanding Ramification Points regarding ramification points on a Riemann surfaces. I'm trying to get a better understanding by altering the ...
2
votes
0answers
13 views

contracted products of torsors

I have a few questions about contracted products of torsors: Is $(A \times^B C) \times^D E \cong A \times^B (C \times^D E) $? Is $A \times^B B \cong A$?
0
votes
1answer
20 views

$I(Z(y-x^{2},z-x^{3}))=(y-x^{2},z-x^{3})$

Why is this: $I(Z(y-x^{2},z-x^{3}))=(y-x^{2},z-x^{3})$ "obvious"? I can easily see $Z(y-x^{2},z-x^{3}) = \lbrace (k,k^{2},k^{3}) \vert k \in\mathbb{A}^{1} \rbrace$ but I have some problems when I ...
4
votes
0answers
47 views

Writing the roots of a polynomial with varying coefficients as continuous functions?

Consider the monic polynomial $$p_{\zeta}(z) = z^n + a_{n-1}(\zeta)z^{n-1} + \dots + a_0(\zeta), $$ where the $a_{i}$'s are continuous functions defined over $\mathbb{C}$. As is well known, the ...
4
votes
3answers
79 views

stalk of projective variety in terms of the coordinate ring

Let $X \subseteq \mathbb{P}^n$ be an embedded projective variety over some field $k$ with its corresponding homogeneous coordinate ring $R = k[X_0,\dots,X_n]/I(X)$. Let further $X = \bigcup_{i=0}^n ...
0
votes
0answers
24 views

a question regarding some proof in Klaus Hulek algebraic geometry

I have a problem with the proof of proposition 1.45. This is the p.42 of Klaus Hulek's Elementary Algebraic Geometry. From the last equation, the author concludes that f(V) is contained in W. But I ...
0
votes
0answers
22 views

Common zeros and GCD of polynomials

Facing another algebraic geometry problem: Let $p,q \in T[x,y]$. Prove the set $V(p,q)$ is finite if and only if set $V(GCD(p,q))$ is finite. ($V(p)$ of course meaning the subset of $A^2(T)$ where p ...
1
vote
1answer
16 views

a question regarding the definition of localization of ring

Let S be a mutliplicative system. Then localization of R is defined by an equivalence relation on R x S. The relation is (a,b) ~ (c,d) if there is an s in S such s(ad-bc)=0 Regarding this, I can't ...
2
votes
0answers
56 views

Consequence of the Hodge index theorem

I'm trying to find an example of two divisors $D_{1} \ D_{2}$ on a complex algebraic projective surface $S$ such that: $D_{1}\equiv D_{2}$ where the equivalence relation is the numerical ...
1
vote
1answer
30 views

Acyclic but not flasque sheaf of abelian group?

Is there a sheaf of abelian groups which is acyclic but not flasque? Maybe we can try $0\to \mathcal{F'}\to \mathcal{F}\to \mathcal{F''}\to 0$ where $\mathcal{F',F''}$ are flasque but $\mathcal{F}$ ...
0
votes
1answer
38 views

Defining the map between tangent space in locally ringed space

I had a doubt studying locally ringed space about what is the canonical map between tangent spaces in the case the residue field is different: Let $(f,f^*):(X,O_X) \to (Y,O_Y)$ a morphism of locally ...
-1
votes
0answers
30 views

Relation between elements of a ring and their annihilators

let $(R.m)$ be a local ring and $x,y$ two elements of $R$ and for ideal $I$ of $R$, we have $x$ is in $I$, $ann(x)=ann(y)$ and $x$ is uniqu minimal ideal of $R$, is there any conditions that implies, ...
3
votes
0answers
66 views

Application of the Riemann-Roch theorem

If C is a quadric hyperelliptic curve ($g(C)=3$ and the canonical line bundle is very ample) contained in the two dimensional complex projective space and $K_C$ is the canonical line bundle of ...
0
votes
2answers
30 views

Hypersurface in $\mathbb P^n$ containing a linear subspace of dimension $r \geq n/2$ has singular points

I'm trying to prove that if I have a hypersurface $X = Z(F)$ (where $F \in K[x_0, \dots, x_n]_{d>1}$) which contains a linear subspace of dimension $r \geq n/2$ then there exists singular points on ...
2
votes
0answers
21 views

Different definitions of an affine algebraic set

Hartshorne assumes we are working over an algebraically closed field $k$ and we simply define algebraic sets to be any set of the form $V(I)\subset k^n$, where $I\subset k[X_1,\ldots,X_n]$ is an ...
3
votes
1answer
7 views

Containment of two varieties with a lot of intersection

Given a projective variety $X\subset \mathbb P^n$ and a curve $C\subset \mathbb P^n$, when can I conclude that $C\subset X$, from the fact that $C$ and $X$ have 'many' points in common. I.e., is there ...
2
votes
1answer
44 views

Restriction of sheaf via inclusion induces isomorphism on stalks

Let $i: Z\rightarrow X$ be the inclusion of $Z $ as a subspace of $X $. Let $\mathscr{F}$ be a sheaf on $X$. The restriction of $\mathscr{F}$ to $ Z $ is defined as the sheafification of $U\mapsto ...
1
vote
1answer
48 views

Lefschetz hypersurface theorem

Let $X \subseteq \mathbf{P}^4$ be a hypersurface over $k$ algebraically closed. Why do we have $Pic(X) = Pic(\mathbf{P}^4)$ by the Lefschetz hypersurface theorem? I only see for $\mu_n$ coefficients ...
1
vote
1answer
24 views

non-singular Riemann surface implies irreducible polynomial without connectedness?

Let $$ F(w,z) = \sum_{i=0}^n a_i(z)w^{n-i}$$ be a polynomial in $z,w$. Define a Riemann surface as the set $$\Gamma:= \left\{ (z,w)\in \mathbb C^2 \mid F(z,w)=0 \right\} $$ and call it non-singular if ...
2
votes
0answers
35 views

Category of schemes with flat morphisms

Consider the category whose objects are schemes, and for every two schemes $X$ and $Y$, morphisms $\operatorname{Hom}(X,Y)$ consist of flat morphisms $X\rightarrow Y$, only. Does this category have a ...
0
votes
0answers
30 views

$H^1(\mathbf{F}_q, Pic(\bar{X})) = 0$?

Let $X/\mathbf{F}_q$ be a smooth projective geometrically integral variety. Does it follow that $H^1(\mathbf{F}_q, Pic(\bar{X})) = 0$? Isn't this just Lang-Steinberg? I think Lang-Steinberg gives us ...
1
vote
1answer
26 views

Index intersection of ample divisors

I'm trying to prove that the sum of two ample divisors on a projective complex algebraic surface S is it self an ample divisor. To do this i need to verify that the index intersection between two ...
2
votes
0answers
52 views

Is the maximal ideal of a localization at a prime ideal principal?

Let $X$ be a closed subvariety of $\mathbf P^{n}_{k}$ which is nonsingular in codimension one. Let $Y$ be a subvariety of $X$ of codimension one, let $\eta$ be its generic point. First question: is ...
7
votes
0answers
70 views
+100

Why is there no theory of $G$-ic varieties, for linear algebraic groups $G$?

A toric variety is an algebraic variety $X$ with an embedding $T \hookrightarrow X$ of an algebraic torus $T$ as a dense open set, such that $T$ acts on $X$ and the embedding is equivariant. It ...
2
votes
1answer
47 views

Geometric interpretation of $H^1_{Zar}(X,\mathcal O_X)$

Let $X/k$ be a smooth projective geometrically integral variety, perhaps over $k$ algebraically closed. What is the geometric interpretation of $H^1_{Zar}(X,\mathcal{O}_X)$? Does it have something to ...
3
votes
2answers
68 views

Elementary algebraic geometry

Let $p(z,w)=z^2+w^2-zw+1,$and $Z(p)=\{(z,w)\in\mathbb{C}\times\mathbb{C}|\,p(z,w)=0\}.$ Is this variety irreducible? Is $Z(p)$ a connected subset of $\mathbb{C}\times\mathbb{C}$ ? (in usual topology ...
3
votes
1answer
27 views

Formal schemes vs formal power series

Take $X = \mathbb{A}^1$ and $Y = \{0\}$. I want to take the formal group scheme at $Y \subset X$. This is a locally ringed space, $(Y, \mathcal{O}_{ \hat{X}})$ where $\mathcal{O}_{\hat{X}}$ is the ...
2
votes
1answer
39 views

Euler characteristic of the structure sheaf

I started to study vector bundles on the spaces, then I have my first contact with the instanton bundles, (bundles that are cohomology of the $0 \to \mathcal{O}^{k}_{\mathbb{P}^n}(-1) \to ...
2
votes
0answers
39 views

Where do divisors on curves come from?

I've been learning about Riemann surfaces and the Riemann-Roch theorem, and I've been able to experience some of its great power. However, I'm curious as to what the history behind divisors on curves ...
1
vote
0answers
26 views

On structure sheaf of an affine scheme

I am reading the algebraic geometry notes by Ravi Vakil. When he proves that the structure sheaf on affine scheme is indeed a sheaf (Thm 4.1.2. in his notes), he first proves that it gives a sheaf ...
1
vote
1answer
96 views

A question on elliptic fibration of K3 surfaces.

Let $X$ be a K3 surface and $L$ be a non-trivial nef line bundle with self-intersection $(L,L)=0$. Then it is known that $h^0(X,L)=2$. How can one prove that the map $\phi_{|L|}\rightarrow ...
1
vote
0answers
14 views

What is the relation between modification and blow-up along the base locus?

Let $X \hookrightarrow \mathbb{P}^N$ be a compact sub manifold of dimension $n$. Let $\mathbb{P}(d, N)$ denote the projectivization of degree $d$ homogeneous polynomial on $\mathbb{P}^N$. Each ...
1
vote
1answer
99 views

Some questions about Hartshorne chapter 2 proposition 2.6

In Hartshorne chapter 2 proposition 2.6,Hartshorne shows that there is a fully faithful functor $t:\mathcal{Var}\rightarrow \mathcal{Sch}(k)$ from the category of varieties over $k$ to the category of ...
0
votes
0answers
16 views

The normalization of a product of varieties

Let $X,Y$ reduced complex analytic spaces, $X^{'}$ and $Y^{'}$ the normalizations of $X$ and $Y$, respectively. Let $(X \times Y)^{'}$ the normalization of $X \times Y$. Is true that $(X \times Y)^{'} ...
1
vote
1answer
50 views

short exact sequences of linear algebraic groups and $K$-forms

This is probably a stupid question, but I can't figure it out. Let $G, G', G''$ be linear algebraic groups over a characteristic $0$ field $K$. Say that $G' \in A$ and $G'' \in B$, where $A,B$ are ...
30
votes
1answer
557 views

Has SGA 4$\frac 1 2$ been typeset in TeX?

The title says it all. I've CW'd the question since I'm answering it - this seemed like the best way to get the news out.
0
votes
1answer
27 views

local dimension of irreducible varieties

If $X$ is an irreducible (quasi-)affine variety it is well known that each maximal sequence $C_0 \subsetneq C_1 \subsetneq \dots \subsetneq C_d$ of irreducible closed subsets has the same length $d = ...
0
votes
1answer
39 views

Fibred product of schemes

I'm learning algebraic geometry, and I'm having some difficulties in developing intuition for the fibred product of schemes. I can take schemes to be over a field (but not necessarily separably ...
0
votes
0answers
15 views

An isogeny from a split algebraic torus

Suppose that there is an isogeny (in the category of commutative algebraic groups) from a split algebraic torus to a semi-abelian variety. Does it follows that this semi-abelian variety is also an ...