Let $X$ be an integral curve, $Y$ a smooth curve, and $f : X \to Y$ a finite surjective morphism. Given a vector bundle $E$ on $X$, what is $\operatorname{rk}(f_{*}E)$? Is it true that $$ \operatorname{rk}(f_{*}E)=\operatorname{rk}(E)\deg(f)? $$ If so, could you give me a proof or indicate a...

$\DeclareMathOperator{\mod}{Mod} \def\O{\mathcal{O}} \def\F{\mathcal{F}} \def\G{\mathcal{G}} \DeclareMathOperator{\qcoh}{QCoh}$Let $X$ be a ringed space. We say that an $\O_X$-module $\F$ is (faithfully) flat if the endofunctor $(-)\otimes_{\O_X}\F$ on $\mod(X)$ is (faithfully) exact. It is...

(Note: I am aware about this post, my question is different) Proposition 4.20 in Görtz-Wedhorn states the following: All of the assumptions and assertions are local in $S,X,Y$ so we may assume they are affine (as we do in the proof of this proposition). However, except for injectivity of $g$,...

I have seen in some titles of articles phrase 'motivic category(ies)' but I haven't found definition. It seems to me that it's just some category of motives or more precisely subcategory of (un)stable motivic homotopy category.

How can we show that the ideal $\mathfrak{p}:=(x^2-y,x^3-z) \subset k[x,y,z]$ is prime using the definition of prime ideals in commutative rings? To show the ideal is prime one defines a map $k[x,y,z] \to k[t]$ given by $f(x,y,x)\mapsto f(t,t^2,t^3)$, and shows that $k[x,y,z]/\mathfrak{p} \cong...

Hartshorne Proposition 1.5 states that any closed subset of a Noetherian topological space can be written as a union of finitely many closed irreducible subsets. A Noetherian topological space is a topological space that satisfies the descending chain condition on its closed subsets. I have one...

I am working through Poncelet Porisms and Beyond by V. Dragović and M. Radnović. In Chapter 3 (Section 3.2, "Algebraic curves in the complex projective plane"), I am stuck on an exercise that makes two claims about the intersection of lines with an algebraic curve. Here is the exact quote from...

The following is a rephrasing of Hartshorne Chapter II Exercise 2.3: For any ring $A$, let $A_{\text{red}}$ be the quotient of $A$ by its ideal of nilpotents. If $\mathcal{S}$ is a sheaf of rings on a topological space, let $\mathcal{S}_\text{red}$ be the sheafification of the presheaf $U...

In the following, I am going to compare two correspondences: $\textbf{correspondence between intermediate fields of $L/K$ and subgroups of ${\rm Aut}_K(L)$}$ vs $\textbf{correspondence between ideals of $R=k[x_1,\cdots,x_n]$ and algebraic subsets of $\mathbb{A}^n_k$}.$ While doing this, I will...

I am self studying Algebraic geometry from Clader and Ross Beginning in Algebraic Geometry and I am struck on Proposition $9.50$ on page $270$. Background: For each $i \in ${$0,1...,n$}, the $i$ th affine patch of $\mathbb{P}^n$ is the set $\mathbb{A}_i^n=${$[a_0,a_1,...,a_n] \in...

Let $A$ be a commutative ring and $M$ an $A$-module. Suppose $M$ is free of rank $n$ with basis $\{ x_1, \ldots , x_n \}$. The $A$-module homomorphism $f \colon M \to A[X_1, \ldots, X_n]$ given by $x_i \mapsto X_i$ induces a unique $A$-algebra homomorphism $F \colon \operatorname{TS}_A(M) \to...

I am self studying Algebraic Geometry from the textbook of Clader and Ross and I have question in Propositions $8.4$ on page $217-218$. Proposition $8.4$: Let $X\subseteq \mathbb{A}^m $ and $Y\subseteq \mathbb{A}^n$ be affine varieties. Then we have $I(X\times Y)= \langle I(X)\rangle+\langle...

I have been self studying the textbook: Beginning in Algebraic Geometry by Clader and Ross and found this book very wonderful simply because most of the proofs are proved in the textbook itself( I donot have a help in real life) and it has a good number of exercises. A lot of intuition is also...

I have been self studying Algebraic Geometry from Clader and Ross's Beginning in Algebraic Geometry and I have a question on Page $195$: Proposition $7.8$. Background information: If $f \in K[x_1,...,x_n]$ and $a=(a_1,...,a_n)\in \mathbb{A}^n$ , then the linearization of $f$ at $a$ is defined by...

I am self studying algebraic Geometry from the textbook of Clader and Ross Beginning in Algebraic Geometry and I have $2$ in the proof of the theorem $7.18$ given on the page $201-203$. Background: define $I_a= \{F\in K[X]\mid F(a)=0\} \subseteq K[X]$. On page $201$ we see that $I_a$ can be...

I was self studying Algebraic Geometry from the textbook of Algebraic Geometry by Clader and Ross: Beginning in Algebraic Geometry. I have questions on page $206-207$ of the textbook. Background information: If $f \in K[x_1,...,x_n]$ and $a=(a_1,...,a_n)\in \mathbb{A}^n$ , then the linearization...

I am reading "The Rising Sea" by Ravi Vakil. I have several question regarding section 3.2 . We know $\operatorname{Spec}\mathbb{C}[x]$ looks like, and we can associate each maximal ideal to a complex number. But what about genric point? What is the intuition behind the terminology: the elements...

Let $E$ be a vector bundle of rank $2$ on $\mathbb P^2$ and $Z$ be zero dimensional subscheme of length $k$ supported on a point $p \in \mathbb P^2$. What is the number $h^0(E(m) \times \mathscr O_Z)$ for an integer $m$? My intuition is that it should be $2k$. But it seems in some literature it...

Let $S=k[x_0,x_1;y_0,y_1]$ be the bihomogeneous coordinate ring of $\mathbb{P}^1\times\mathbb{P}^1$. Suppose that $F\in S_{(d_1,d_2)}$ is a generic bihomogeneous form of bidegree $(d_1,d_2)$, and let $F_1\in S_{(a_1,b_1)}$, $F_2\in S_{(a_2,b_2)}$ be generic forms of lower bidegrees. The question...

I'm reading CLS's Toric Varieties right now, and something is confusing me. By Theorem 1.1.17, an affine variety $V$ is toric iff $I(V)$ is toric, i.e. prime and generated by binomials. Now the variety $V = V(X+Y-Z) \subset \mathbb{C}^3$ seems to me to be toric simply because it's isomorphic to...