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Look at the affine pieces: over the open subset $u \neq 0$, you have a local coordinate $z = v/u$ and your equations can be written as $y = zx$ and $xy = x^6 + y^6$. Substituting $y$ in the second equ …

On a complete nonsingular curve over an algebraically closed field, a line bundle of degree zero with a global section is necessarily the trivial bundle. This is lemma IV.1.2 in Hartshorne's Algebraic …

This is not a complete answer by any means, but here are the two most basic arguments. First of all, you have that every projective scheme that can be embedded in $\mathbb{P}^n$ can be covered by $n+1 …

The Kähler cone of a del Pezzo surface of degree 6 is not simplicial: see section 6 of these notes.

Let $G$ be an affine algebraic group over $\mathbb{C}$. According to SGA3, any closed normal subgroup $N$ is representable by an affine algebraic group, as is the quotient $G/N$. These statements ar …

Consider the exact sequence $0 \to S(-\mathrm{deg}\; f) \to S \to S/(f) \to 0$ (where the first map is multiplication by $f$) and take its long exact sequence of $\mathrm{Ext}$ groups. Since both $S$ …

UPDATE: My answer essentially just gives the definition of Kahler differentials and differential forms and misses the point of the question. Georges' answer addresses the relationship between the two. …

The first example is the twisted cubic in $\mathbb{P}^3$.

Actually, there are two different restriction maps: The first one (the one you correctly say is neither surjective nor injective in general) is that on sections: for $\mathcal{F}$ a sheaf on a schem …

As Kevin said, the higher cohomology groups of constant sheaves on irreducible varieties are zero when working with the Zariski topology. Also, "fibre bundles aren't locally trivial" and "the inverse …

You say that a ring homomorphism $\phi: A \to B$ is étale (resp. smooth, unramified), or that $B$ is étale (resp. smooth, unramified) over $A$ is the following two conditions are satisfied: $A \to B …

It might not be the best reference for a systematic study of stacks and some of the terminology is old, but Mumford's "Picard Groups of Moduli Problems" (1965) might be a nice complement. It explains …

The bare question: Let $\mathcal{C}$ be an $\infty$-topos, and let $\tau_{\leq 0}\mathcal{C}$ be the subcategory of 0-truncated objects (which is the nerve of an ordinary Grothendieck topos: see HTT 6 …

There is a nice treatment of it in chapter 1 of Brylisnki's Loop Spaces, Characteristic Classes and Geometric Quantization. In the Stacks projects, look for section 19.19.

Here's a basic (and often used!) example: the zero locus of a homogeneous polynomial of degree $d$ in $\mathbb{P}^n$. For concreteness, let me spell out the case $n = 1$, $d = 2$. Cover $\mathbb{P}^1$ …