The introductions of each chapter of Geometry of Algebraic Curves, Volume II, Arbarello Enrico, Cornalba Maurizio, Griffiths Phillip (the book itself is tuff, but the introductions are extremely ...

Abotu the first question: yes, it does. Already for $\mathbb{P}^n$ embedded with $\mathcal{O}(d)$, the cone changes with $d$. About the second, you might view a cone as a variety with a $\mathbb{G}_m$...

The square of the hypotenuse is equal to the sum of the squares of the other two sides.

Let us discuss first the differntial geometry case. Here, you can use the tubular neighbourhood theorem to extend the vector bundle. Explicitly, you have a neighbourhood $U$ of $Y$ in $X$ which is ...

Some basic references are: the book by Manin Gauge Field Theory and Complex Geometry ; the first paper by Delinge in Quantum Fields and Strings: A Course for Mathematicians ; the recent paper on the ...

First, glue two points, obtaining a singular curve of genus $g_1+g_2$. Then smooth out the singularity. Equivalently, remove two discs, one from each curves, and glue two small annuli around the discs ...

I like the book by Claudio Procesi Lie Groups: An Approach through Invariants and Representations

The answer might be in Michel Demazure. Automorphismes et déformations des variétés de Borel. Invent. Math., 39(2):179–186, 1977

Yes! This is sometime called naturality of Chern classes. You can find it in many books, for instance Complex Geometry - An Introduction | Daniel Huybrechts, or Differential forms in algebraic ...

Yes! It follows from the fact that $$ d\pi \circ d\theta=d\pi $$ whihc in turn follows from $\pi\circ \theta=\pi$. I think you actually have an equality $H^2(X,\mathbb{Z})^{\theta}=\pi^*H^2(Y,\mathbb{...

See for instance Manin, Yuri I. Gauge field theory and complex geometry or Quantum Fields and Strings: A Course for Mathematicians

This should be true: let $X=Spec(B)$ and $Y=Spec(A)$ be Noetherian affine schemes, if $$ f\colon X\to Y $$ is faithfully flat and $X$ is normal, then $Y$ is normal. A proof is given in Corollary 15.4 ...

It should be representable by an ind-scheme. You should look at the papere by Mumford "On the equations defining abelian varieties II", section 9. He calls this moduli space $\mathcal{M}_{\infty}$, it ...

To start with: Galois Groups and Fundamental Groups, by Tamás Szamuely (The theorem in section 3.4 about the absolute Galois group go $\mathbb{C}(t)$ is amazing and enlightening.) And then: Milne, ...

The best (algebraic) advice I can give is to look at section 2.2.5 of https://arxiv.org/pdf/1812.03538.pdf ( Uniqueness of K-polystable degenerations of Fano varieties by Blum and Xu) (eventually, ...

I have received the following great answer from Vlad Lazić: In the numerical setup, what you ask (with some assumptions on the base, such as Q-factoriality) is well-understood, see the paper of ...