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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Computing the dualizing sheaf of an exceptional divisor

Let $D_1=Z(f_1)$ and $D_2=Z(f_2)$ be two plane cubics meeting in nine points $p_1,\dots,p_9\in\mathbb P^2$. The $f_i$'s induce a rational map $$(f_1,f_2):\mathbb P^2\dashrightarrow\mathbb P^1$$ ...
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$J \subset I(V(J))$ where $J$ is an ideal.

The textbook says it's by definition, but as I see it the inclusion should be reversed should it not? I mean $I(V(J))= \{ f\in k[x_1,\dots, x_n]: f(a_1,\dots ,a_n)=0 \text{ for an arbitrary element } ...
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18 views

Alternative proof of Noether Normalization Lemma

On Mumford's Red Book (end of section 7 of chapter 2, pg 126, 127), there is an alternative proof of Noether's Normalization Lemma that goes like this: For an affine variety $X$ over an algebraically ...
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16 views

What does it mean to restrict a function germ to a set germ?

Two sets $S$ and $T$ define the same germ at a point $\xi$ in a topological space $M$ if there is a neighbourhood $U$ of $\xi$ such that $S \cap U = T \cap U$. Two functions $f,g : M \rightarrow \Bbb ...
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Identifying lines in $\mathbb P^2$

Let $L$ and $M$ be two lines in $\mathbb P^2$. Does there exist a map $f : \mathbb P^2 \to X$ that "identifies" $L$ and $M$, in the sense that $f\vert \mathbb P^2 \setminus (L \cup M)$ is an ...
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47 views

Euler characteristic, genus and cohomology: a deep connection?

For a smooth projective curve $V$ over the complex numbers, the algebraic genus, defined as the dimension of the linear system $L(\omega)$, where $\omega$ is the canonical divisor, coincides with the ...
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17 views

Tensor product of local Artinian rings

Consider a complete Noetherian local ring $R$ and two local Artinian $R$-Algebras $A$ and $B$. I'm trying to prove that the spectrum $\text{Spec}(A\otimes_{R}B)$ is connected or, equivalently, that ...
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212 views

Why can we use flabby sheaves to define cohomology?

In my algebraic geometry class, we defined sheaf cohomology using flabby sheaves, and the functor on the category of sheaves on a space $X$: $$ D: \mathcal F \mapsto D\mathcal F $$ where $$ ...
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27 views

27 lines on Fermat cubic

Fermat cubic is $S=\{(x:y:z:w)|x^3+y^3+z^3+w^3 =0\} \in \mathbb{P}^3$. It is obvious that 27 lines on Fermat cubic are represented by $(x,ax,z,bz)$ for cube root $a,b$ of $-1$ and their conjugates. ...
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31 views

Very ample divisors and the Riemann-Roch theorem

What is the easiest way to prove that a divisor $D$ is very ample if and only if $l(D - P - Q) = l(D) - 2$ for all points $P, Q \in C$. It must be a consequence of the Riemann-Roch theorem, but I am ...
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28 views

An algebraic curve $C$ as a cover of $\mathbb{P}^1$ over $\bar{\mathbb{Q}}$

Could someone explain what it means for an algebraic curve $C$ to be a cover of $\mathbb{P}^1$ over $\bar{\mathbb{Q}}$, ramified over $n$ points?
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18 views

singular point of a complete intersection surface

Let $S:= H_1\bigcap H_2\bigcap \cdots \bigcap H_N \subset\mathbb{P} _{\mathbb{C}}^{N+2}$ be a complete intersection surface, where each $H_i$ is a hypersurface defined by a homogeneous equation $f_i$. ...
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1answer
32 views

Subjects or recent progress in Tropical geometry or similar suitable for undergraduate investigation

I'm worried this might not be fitting for this forum, but it's basically a literature and reference request. I'm looking to do a project in algebra where we are supposed to research some topic ...
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18 views

Invariants of $O(2) \times O(2)$ under simultaneous conjugation

Let $G= \textrm{O}(2)$ be the group of orthogonal $2 \times 2$ matrices over $\mathbb{C}$. $G$ acts on $G \times G$ by conjugation: $g \cdot (a,b) :=(g a g^{T}, g b g^T)$. This induces an action on ...
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28 views

non-abelian Galois cohomology

Let $1 \to A \to B \to C \to 1$ be a short exact sequence of (not necessarily abelian) $G$-modules. Passing to non-abelian cohomology, we have the exact sequence of pointed sets $$ 1 \to A^G \to B^G ...
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1answer
63 views

About Plücker embedding

I'm doing a work about Plücker embedding and I need some help about a few topics. I'm going to list them: $1-$ I know that Plücker embedding is well-defined and is injective. However, Plücker ...
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44 views

What is the canonical morphism of $\mathbb{P}^n_A$ to $\text{Spec }A$?

On Liu's book Algebraic Geometry and Arithmetic Curves, he gives a general definition of the projective scheme $\mathbb{P}^n_A$, for $A$ a ring, as $Proj (B)$, for $B=A[x_0,\dots,x_n]$. Later, he ...
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1answer
45 views

Proof of Nike's trick: Two affine open subsets contain a simultaneously distinguished open subset

I'm trying to work through this proof of Nike's tick. Statement of the lemma: Let $ U_{i} = Spec\ A_{i} $ for $ i\in\{1,2\} $ be two open affine subschemes of a scheme $ X $. For $ x\in U_{1}\cap ...
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40 views

Is the image of a morphism between affine schemes always constructible?

Is there example for $f\colon A\to B$ being ring map, but the image $f^*\colon \operatorname{Spec}(B)\to \operatorname{Spec}(A)$ not constructible? (i.e., written as a finite union of locally closed ...
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1answer
43 views

$I$-smoothness in Algebraic Geometry

I was reading in Chapter 10 of Matsumura's book about $I$-smoothness. In the book, the autor defines this concept by the following universal property: Let $A$ be a ring, $B$ an $A$-algebra and $I ...
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17 views

Polynomial Ring of Linear Algebraic Group

During lectures, we defined the Linear Algebraic group as the algebraic set $ GL(V):=k^{n^2}-V(Det) $ Where $V(Det)$ are the matrices with $0$ determinant. Then we proceed by identifying the ...
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39 views

Geometric Meaning of Luna's Slice Theorem

I found the Luna's Slice Theorem very Technical. It will be helpful if someone illustrates the geometry involved in the theorem. Also why this theorem so useful? This is Luna's Slice theorem from a ...
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1answer
32 views

Global sections of the projective space

Let $k$ be an algebraically closed field, and let $\mathbb{P}^n_k=\operatorname{Proj}(k[x_0,x_1,\dots,x_n])$, with structure sheaf $\mathcal{O}$. I would like to know how to prove that ...
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1answer
52 views

Is (0,0) of $V(x-y^2)$ a smooth point?

I'm pretty sure it is a smooth point since given $f(x,y)=x-y^2$ the gradient $df=(1,-2y)$ is always non-singular. I'm asking because page 22 of Principle of Algebraic Geometry says: ...
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44 views

Some questions about Qing Liu's proof of Valuative Criterion of Properness

I'm studying algebraic geometry by following Mumford's Red Book, but recently I found Qing Liu's book "Algebraic Geometry and Arithmetic Curves", which seems to be more careful and detailed in its ...
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26 views

is configuration space an H-space?

Let $X$ be a manifold. Let $F(X,n)$ be the configuration space of order $n$. Let $B(X,n)=F(X,n)/\Sigma_n$ be the unordered configuration space of order $n$. Is $B(X,n)$ an H-space? Under what ...
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1answer
52 views

Chapter dependency tree for Hartshorne's Algebraic Geometry

I'm self-studying Hartshorne's Algebraic Geometry and I need some guidance. I've studied chapter I (varieties) and sections 1, 2 and 3 of chapter II (schemes). Do I need to study all sections in ...
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23 views

Are $k[Z_1\sqcup Z_2]\cong k[Z_1]\times k[Z_2]$ as coordinate rings?

Suppose $X$ is an affine variety over algebraically closed $k$. If you can decompose $X$ as a disjoint union $X=Z_1\sqcup Z_2$ for each $Z_i$ closed, is there some relationship between the coordinate ...
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36 views

About images of (prime) ideals under injective endomorphisms

Let $f : R \to R$ be an injective unitary endomorphism of a commutative ring with 1. Let $I$ be an ideal of $R$. I have several related questions concerning the image of $I$ under $f$: 1) Under which ...
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1answer
32 views

closed and open subscheme of affine scheme

Let $X=Spec(A)$ be a noetherian affine scheme. Let $I_1, \ldots, I_n$ be ideals of $A$ such that $I_i + I_j = 1$ for all $i \neq j$. Define $X_i = Spec(A/I_i)$ so that X is the disjoint union of the ...
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30 views

A question about varieties (proposition from Miles Reid undergraduate comm algebra)

In Miles Reid, undergraduate commutative algebra, I read the following: "Suppose that $k$ is an algebraically closed field and that $A=k[x_1,...,x_n]$ is a finitely generated $k$-algebra of form ...
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1answer
34 views

A natural map of higher direct images.

Let $f\colon X \rightarrow Y$ be a proper morphism of schemes, $Y$ noetherian (I don't know which of those assumptions is actually needed for the claim). According to my lecture notes, for ...
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28 views

Closed subscheme vs open subscheme in ring of dual numbers

If we take the ring of dual numbers $R=k[x]/(x^2)$ for algebraically closed field $k$, we note that by the ideal correspondence theorem, the only prime ideal in $R$ is $(x)$. Thus the scheme $spec ...
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34 views

History of a result from Bézout

BÉZOUT'S THEOREM: Let $F$ and $G$ be projective plane curves of degree $m$ and $n$ respectively. Assume $F$ and $G$ have no common component. Then $\displaystyle\sum_{P}I(P,F\cap G)=mn$ $I(P,F\cap ...
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33 views

Dimension of irreducible variety

Why is the dimension of intersection, $V\cap H$, of $m$-dimensional irreducible variety $V$ and a hyperplane given by $\dim(V\cap H)$ of dimension $m-1$?
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31 views

Question about composition of morphisms of schemes (Mumford's)

On section 7 of Chapter 2 of Mumford's Red Book, there is the following statement: Suppose $X\stackrel{f}{\rightarrow}Y\stackrel{g}{\rightarrow}Z$ are morphisms of schemes, such that $g$ is of finite ...
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1answer
52 views

Cohomology groups of coherent sheaves for very small and very big twists.

Let $\mathcal{F}$ be nonzero coherent sheaf over the projective space $\mathbb{P}_k^n$. The Serre vanishing Theorem says that $h^i \mathcal{F}(d)=0$ for $i>0$ and $d\gg 0$. I am wondering if it is ...
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48 views

Is the Zariski Topology

if $ K $ is an algebraically closed field, asks: Is there a point $ "w" $ of $ K ^ n $, is closed in the Zariski toplogy?
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1answer
45 views

Is a torsor over a variety a variety?

Let $X$ be an algebraic variety over some field $k$ of characteristic 0. Let $g : Y \to X$ be a $X$-torsor under some linear algebraic $k$-group $G$. Is $Y$ also an algebraic variety over $k$?
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28 views

Galois Group acting on disjoint union of Schemes

Given a finite Galois fiedl extension $L/K$ with Galoisgroup $G$ i.e. the fixpoints of $L$ under G are $K$ ($L^G=K$). Let $G$ act on $L^n$ semilinearly (i.e. $g(ax)=g(a)g(x)$ where $a\in L$ and $x\in ...
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45 views

When is a holomorphy ring a PID?

I will use the notation and language of Stichtenoth, Algebraic Function Fields and Codes. Let $F$ be a function field over a finite field $\mathbb F_q$, $S$ a non empty set of places (possibly ...
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44 views

Is quotient under $S_4$ action on “cube” representation a flat morphism?

Consider a three-dimensional irreducible representation $V$ of $S_4$, corresponding to symmetries of cube. Let $p$ be canonical projection $p: V \rightarrow V/S_4$. My question: is $p$ flat? I want ...
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72 views

Does the category of locally ringed spaces have products?

The category of schemes has all fibered products, but the proof uses affine schemes in a crucial way. I want to understand whether this is true for the category of locally ringed spaces. The standard ...
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9 views

A proof about resultant and common zero in Klaus Hulek book

Here in the proof of lemma 5.6, it says the equality qr = ls implies that qr, ls have same zeros. Then how to conclude that r and s have same zeros? I think q, s can have same zeros or r, l can ...
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17 views

Question about affine coordinate changes

Fulton in his book defines affine coordinate changes: I'm trying to prove the item (b) of this question: Let's prove using the induction suggestion. Suppose $V=V(F_1)$, where ...
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1answer
30 views

Is a morphism between smooth varieties smooth if fibers are?

Suppose that $X$ and $Y$ are smooth varieties over a field $k$ (not necessarily algebraically closed), of dimension $m$ and $n$. Suppose we are given a morphism $\pi:X\rightarrow Y$. We know that if ...
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1answer
34 views

Show that something is a subsystem of a complete linear system

I have a simple and basic question concerning degree of projective curves and I'm referring to something I've read on Miranda's book, Algebraic curves and Riemann Surfaces, Chapter VII, 3. The Degree ...
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19 views

If $V$ is a quasi-affine variety, the algebra $k[V]$ is isomorphic to a subalgebra of a finitely generated $k$-algebra?

Suppose $V$ is a quasi-affine variety over an algebraically closed field $k$. I read that the algebra $k[V]$ is isomorphic to a subalgebra of some finitely generated $k$-algebra. However, I thought ...
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61 views

Is $\mathbb{H}P^\infty$ an H-space or not?

$\mathbb{R}P^\infty$ is H-space. $\mathbb{C}P^\infty$ is H-space. Is $\mathbb{H}P^\infty$ an H-space or not?
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23 views

On different versions of Schwarz Zippel

Theorem (Schwartz, Zippel). Let $P\in F[x_1,x_2,\ldots,x_n]$ be a non-zero polynomial of total degree $d≥0$ over a Field $F$. Let $S$ be a finite subset of $F$ and let $r_1,r_2,...,r_n$ be selected at ...