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Questions tagged [algebraic-geometry]

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. This tag should not be used for elementary problems which involve both algebra and geometry.

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Proving representibility of some functor in Gortz's Algebraic Geometry book.

I want to understand proof of next proposition ( Gortz's Algebraic Geometry, Proposition 8.4. ) Proposition8.4. Let $S$ be a scheme and let $v : \mathcal{E} \to \mathcal{F}$ be a homomorphism of ...
0 votes
1 answer
49 views

The inverse image functor induced from morphism of ringed spaces commutes with the image?

Let $ f: (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces and let $ w : \mathcal{F} \to \mathcal{F}'$ be a homomorphism of $\mathcal{O}_Y$-modules. Then Q. $f^*(\...
1 vote
0 answers
35 views

Scheme-theoretic construction of tensor product of vector bundles on a scheme.

Let $X$ be a scheme and let $\mathcal{E}$ and $\mathcal{F}$ be locally free sheaves of finite rank (maybe works with coherent sheaves?). Let $E = \mathrm{Spec}(\mathrm{Sym}(\mathcal{E}^\vee))$ denote ...
0 votes
0 answers
9 views

Closed fibre of the exceptional divisor of a smooth blow up, Harthshorne II 8.24

Let $X$ be a smooth variety and $Y\subset X$ is a smooth subvariety. We blow up $X$ along $Y$ to get $\tilde X\to X$ with exceptional divisor $E$. Let $p$ be a closed point of $Y$ with fibre $Z=p\...
0 votes
0 answers
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Confusion on proof of "dimension of the fiber is an upper semicontinuous function on the source" (From Vakil FOAG, Theorem 12.4.3).

I'm confused on a step of the proof of Theorem 12.4.3(a) in Vakil's FOAG. The theorem states: Suppose $\pi: X \rightarrow Y$ is a morphism of finite type $k$-schemes. Then the dimension of the fiber ...
1 vote
1 answer
47 views

Hilbert polynomial of the union of two intersecting lines

I am trying to compute the Hilbert polynomial of the union of two intersecting lines $L_1$ and $L_2$ in $\mathbb{P}^3$. In order to get it, I supposed WLOG that $L_1=Z(x,y)$ and $L_2=Z(y,z)$. Thus, ...
1 vote
1 answer
55 views

Correspondence between sheaves of ideals and closed immersion

Let $(X,\mathcal O_X)$ be a locally ringed space. Let $\mathcal J$ be a sheaf of ideals over $\mathcal O_X$. Then we can construct a closed immersion of locally ringed spaces: $$(Z(\mathcal J),i^{-1}(\...
0 votes
0 answers
66 views

Fundamental group of a smooth projective surface over $\mathbb{R}$.

Let $X_{\mathbb{R}}$ be a smooth, projective surface over $\mathbb{R}$. Then its complexification, $X$, is a smooth complex projective surface. $X_\mathbb{R}$ is a manifold, so we can talk about its ...
0 votes
0 answers
35 views

Finite covering of Hirzebruch surfaces

Question Let $F_n = \mathbb{P}(\mathscr{O}_{\mathbb{P}_1}\oplus\mathscr{O}_{\mathbb{P}_1}(n))$ be an Hirzebruch surface and consider its finite covering(or double covering) $f: X \to F_n$. Let $f$ be ...
1 vote
0 answers
42 views

Generic hyperplane sections of an irreducible projective variety is irreducible

I am reading Principles of Algebraic Geometry by Griffiths and Harris. They asserted that the generic hyperplane section $H\cap V$ of an irreducible nondegenerate variety $V$ of dimension $\geq 2$ is ...
0 votes
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Equivalent ways to define structure sheaf on $Spec(A)$.

How do we show that the following two ways of defining the structure sheaf on $Spec(A)$ are the same? Definition 1 $$\mathcal{O}(U) = \left\{s \in \prod_{\mathfrak{p} \in U} A_{\mathfrak{p}} : s \text{...
16 votes
3 answers
3k views

Geometric reason why elliptic curve group law is associative

The question title says it all. I am looking for a geometric proof for the fact that the group law defined on elliptic curves is associative. I've heard somewhere about something on the internet ...
2 votes
1 answer
41 views

The Lie algebra of an algebraic group

I have a question concerning different visions of the Lie algebra of an algebraic group. To be more specific, let $G = \operatorname{Spec}(H)$ be a smooth (finitely generated) algebraic group over a ...
-1 votes
0 answers
29 views

How to define the action of $\operatorname{Aut}(X_i/S)^\circ$ on $A$?

I'm reading Lei Fu's "Etale Cohomology Theory". For the sentence highlighten, my question are: (1) How to get $\operatorname{Aut}(X_i/S)^\circ$? (2) How to define the action of $\...
1 vote
0 answers
72 views

Smooth algebraic plane curves are open in Zariski [duplicate]

Homogeneous polynomials of degree $n$ can be seen as points of $K^m\setminus\{0\}$. It's natural to see plane algebraic curves of degree $n$ as points of the projective space. I was told that smooth ...
4 votes
0 answers
64 views

Quotient of Special Linear Group by its centre

I am currently writing my Master's Thesis about Moduli Spaces of Groups Representations via the GIT quotient, i.e. in the context of Algebraic Geometry. Hence, algebraic groups are fundamental objects ...
-1 votes
0 answers
38 views

Endomorphisms of $\mathbb{G}_a$ given by polynomials in frobenius [closed]

I have heard the statement that the endomorphisms of $\mathbb{G}_a$ as a group scheme are given by polynomials in the frobenii, but have not been able to find a proof. I know that endomorphisms of $\...
1 vote
0 answers
32 views

Closed immersions into affine scheme

Let $(\operatorname{Spec}A,\mathcal O_A)$ be an affine scheme, for a ring $A$. I'm trying to prove that the closed subschemes of $(\operatorname{Spec}A,\mathcal O_A)$ are exactly those of the form $(V(...
13 votes
4 answers
2k views

Quasiseparated if finitely covered by affines in appropriate way

I've been reading Vakil's notes on algebraic geometry (on my own -- this is not part of a class), and I'm stuck on one problem (number 6.1.H). It goes as follows. Let $X$ be a scheme. Prove that $X$...
6 votes
2 answers
3k views

Stalks and direct image

Let $f: X \rightarrow Y$ be a continuous map of topological spaces, and $F$ a sheaf of rings on $X$. The direct image sheaf $f_{\ast}F$ on $Y$ is given by the formula $V \mapsto F(f^{-1}V)$. If $x \...
5 votes
0 answers
67 views

Generic bound on quadratic character sum

Let $\chi$ be the non-trivial quadratic character of $\mathbb{F}_q$, and let $f(x)$ be a square-free polynomial over $\mathbb{F}_q[x]$. Then by the Weil bound, we have the generic estimate $|\sum_{x\...
0 votes
0 answers
45 views

Accessible result motivating Arithmetic Geometry

My understanding of arithmetic geometry is that it applies concepts originating from algebraic geometry to schemes that are quite different from these initially studied in this field (those associated ...
4 votes
1 answer
954 views

Global sections on quasi coherent sheaves on affine scheme

This is a lemma from Hartshorne's Algebraic Geometry. Let $X=\text{Spec}(A)$ be an affine scheme $f\in A, D(f)\subseteq X$. Let $\mathcal{F}$ be a quasi coherent sheaf on $X$. If $s\in \Gamma(X,\...
1 vote
1 answer
51 views

Isomorphism $\mathbb{A}^2 \to \mathbb{A}^2$

This exercise is taken from the notes of Gathmann of Algebraic Geometry, in particular it is the point d) of the exercise 4.13 of the version 2021/2022. If $f:\mathbb{A}^2 \to \mathbb{A}^2$ is an ...
1 vote
1 answer
26 views

the infinite category of pullback squares in an infinite stable category is also stable

I am currently reading Lurie's paper infinite stable category, in the proof of proposition 4.4, to show that every pushout square in an infinite stable category $C$ is also a pullback, he considers ...
1 vote
1 answer
97 views

Infinite category structure on SCRing and 'space of commutative squares' in SCRing

When I read this paper 'virtual cartier divisors and blow ups', I often meet with such phrase like 'mapping space of infinite category $SCRing_{A}$'. See lemma 2.3.5 in the above paper: $Map_{SCRing_{...
0 votes
0 answers
35 views

Can an ample line bundle equipped with a metric which is negative somewhere?

Let $X$ be a compact complex manifold and $L$ a holomorphic line bundle on $X$. As is wellknown, $L$ is ample if and only if $L$ admits a positive Hermitian metric (i.e., its curvature form is ...
1 vote
1 answer
683 views

Doubts on the proof about the classification of irreducible projective conics

There are some details in the proof of corollary 3.12 on Frances Kirwan's book complex algebraic curves that I can't understand. Corollary 3.12 Any irreducible projective conic $C$ in $P_2$ is ...
0 votes
0 answers
48 views

Proposition I-3.5 in Hartshorne

The following is the statement of Proposition I-3.5 in Hartshorne. Let $X$ be any variety and let $Y$ be an affine varietiy. Then there is a natural bijective mapping of sets$$\alpha:\text{Hom}(X,Y)\...
8 votes
1 answer
2k views

Maximum number of singular points on irreducible curve in $\mathbb{CP}^{2}$

Let $C$ be a degree $d$ irreducible curve in $\mathbb{CP}^{2}$. Can we find maximum number of singular points in $C$? For $d=2$, I find that there is no singular point on irreducible conic. (If there ...
1 vote
0 answers
33 views

Suggestions for references about filtrations and their topological properties

I've started reading Qing Liu's Algebraic Geometry, and whilst the prerequisites on multilinear algebra and flat modules were familiar, I've hit a wall with his discussion of the topology endowed on a ...
0 votes
1 answer
29 views

Coordinate ring of an open affine variety?

I have trouble understanding what the coordinate ring of an open affine variety, i.e. the ring of regular functions on that variety, is (in the classical sense). Let $k$ be algebraically closed. If $X ...
2 votes
1 answer
73 views

Details on: Flag incidence variety is a projective variety

Let $0 \leq p \leq q \leq n$. Define the flag incidence variety $$\text{Fl}(p,q,n):=\{(V,W) : V \leq W, V \in \text{G}(p,n), W \in \text{G}(q,n)\}$$ where $\mathrm{G}(i,n)$ is the Grassmannian, i.e. ...
0 votes
0 answers
33 views

When is the diagonal morphism regular?

Let $X$ denote a smooth algebraic variety over a field $k$. When is the diagonal map $X \to X \times_{k} X$ regular? This question is somewhat related to Diagonal morphism of regular variety is a ...
0 votes
0 answers
54 views

Do étale coordinates give rise to a regular sequence of diagonal elements?

Fix an algebraic extension $k\subseteq K$ of fields of characteristic zero and consider a map of commutative rings $\phi\colon K\left[T_{1}^{\pm},\dots,T_{n}^{\pm}\right]\to A$ which is étale. Now ...
2 votes
1 answer
76 views

Can shapes e.g. triangles and circles, be added?

Addition, subtraction etc. seem like basic operations we try/test often. My son wondered if shapes can be added. My guess is that they can be e.g. if we convert shapes to algebraic representation, we ...
0 votes
2 answers
47 views

Projective varieties contained in dense open subsets

Let $X$ be a smooth irreducible projective variety over the complex numbers. Let $U$ be a nontrivial dense open subset of $X$. Does there exist a projective curve $C$ inside $U$? My attempt: Let's ...
1 vote
0 answers
39 views

When is the long exact sequence of sheaf cohomology a series of many short exact sequences

Let $0\rightarrow \mathcal A \rightarrow \mathcal B \rightarrow \mathcal C \rightarrow 0$ be an exact sequence of sheaves of $\mathcal O_X$-modules on a (say projective, normal) scheme $X$. Taking ...
-2 votes
1 answer
92 views

Understanding the consequences of Hilbert Nullstellensatz

So I was reading Fulton's Algebraic Curves and was stuck in Hilbert's Nullstellensatz, specifically, its direct consequences (see photo). Corollary 1-3 Corollary 1. If $I$ is a radical ideal in $k[...
1 vote
1 answer
143 views

Two-dimensional unipotent algebraic groups

How to prove that two-dimensional unipotent algebraic group over the field of characteristic 0 is commutative?
2 votes
1 answer
88 views

Homogeneous spaces for unipotent groups are isomorphic to $\mathbb{A}^n$

I am looking for a proof of the following fact: Let $U$ be a unipotent algebraic group over $\mathbb{C}$. Then any $U$-homogeneous space is isomorphic to $\mathbb{A}^n$. Context: Let $G$ be a ...
0 votes
1 answer
39 views

Self intersection number of diagonal

Suppose I have an elliptic curve $E$. How would I calculate the self intersection in $A_*(E \times E)$ of the diagonal $\Delta$? It seems the formula I need to use is $\Delta . \Delta = c_1(N_{\Delta/...
3 votes
1 answer
159 views

How do I interpret the intersection of a variety with a "non-closed hyperplane?"

I am trying to understand Vakil's statement and proof of Bertini's theorem, which has been updated since many of the questions related to it were posted on this website (for what it's worth, I'm not ...
1 vote
2 answers
336 views

Prove that an isolated point of $C: (f=0)\subset \mathbb{R}^2$ must be a max or min of $f: \mathbb{R}^2 \rightarrow \mathbb{R}$.

Let $f\in \mathbb{R}[x,y]$ and let $C: (f=0)\subset \mathbb{R}^2$; we say that $P\in C$ is isolated if there is an $\epsilon >0$ such that $C\cap B(P,\epsilon)=P$. Prove that if $P\in C$ is an ...
0 votes
0 answers
77 views

Polynomials in $k[x,y]$ where $f(x,y) = 0$ iff $x=y=0$

If $f(x) = x^n - a$ has no root in $k$, we have that $x^n - ay^n = 0$ iff $x=y=0$. However, there are fields that are not algebraically closed where we can't find a polynomial of the form $x^n - a$ ...
1 vote
1 answer
43 views

Relative differential of vector bundle

I am reading Ogus and Vologodsky's article on nonabelian Hodge theory in characteristic $p$ (https://math.berkeley.edu/~ogus/preprints/anonhodge.pdf). It uses the terminology ''vector groups'' in the ...
2 votes
1 answer
142 views

Projective space as Grassmannian

I want to understand how (or rather, why) the schemes representing the functors of $n-1$-dimensional projective space and the Grassmannian of lines are equivalent. More concretely, for an integer $n &...
2 votes
0 answers
55 views

Line Bundles on Elliptic Curves lift From Quotient

Suppose I have an elliptic curve $E$ and take its quotient by some finite group $G$. Is there some relation between the line bundles on $E$ and the line bundles on $E/G$? For example does every line ...
0 votes
1 answer
74 views

There can't only be non-homogenous prime ideals between two homogenous primes. (Vakil 12.2.G, part c)

I'm stuck and looking for advice on part c of Question 12.2.G of Vakil's FOAG. The question states: (a) Suppose $X \subset \mathbb{P}^n$ is an irreducible projective $k$-variety. Show that the affine ...
4 votes
2 answers
217 views

Calculating the fibres of a scheme morphism are proper but the morphism is not proper

$\newcommand{\Spec}{\operatorname{Spec}}\newcommand{\C}{\mathbb C}\newcommand{\A}{\mathbb A}$ Let $f:\mathbb A^1_\mathbb C\rightarrow \mathbb A^1_\mathbb C$ be induced by the ring homomorphism $t\...

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