I am currently working through Fulton's Algebraic Curves and I have attempted the following problem: $$\text{Let P be a simple (non-singular) point on }F\text{ . Show that the tangent line to }F \text{ at } P\text{ has the equation }F_X (P )X + F_Y (P )Y + F_Z (P )Z = 0$$ My solution thus far...

Let $k$ be a commutative ring and let $A$ be a commutative $k$-algebra. The cotangent complex $\mathbf{L}_{A \mid k}$ can be computed using a simplicial resolution of $A$ as follows: choose a simplicial commutative $k$-algebra $P$ and an augmentation $\epsilon : P_0 \to A$ (i.e. a $k$-algebra...

Vakil, Definition 20.1.1: Let $X$ be a variety (reduced separated finite type scheme, actually I'm not sure if we need all this. I think we can get away with dropping "reduced" and "separated" and just assume $X$ is a finite type scheme) over a field $k$ (not necessarily algebraically closed)....

I am trying to work through the following problem: Let $k$ be a field, and let $X = \operatorname{Spec} k[x, y, z, w]/(xy−zw)\subseteq \mathbb{A}^4_ k.$ (a) Show that $D= V (x, z)$ is a prime (Weil) divisor on $X$ and that $\operatorname{Cl} X\simeq \mathbb{Z}$ is generated by the divisor class...

Many results in real algebraic geometry are proved by passing from a real variety to its complexification , then studying the action of complex conjugation on . For example, one often regards $X(\mathbb R)$ as the fixed-point locus of conjugation on $X(\mathbb C)$. This appears in results such...

If $f \colon X \to Y$ is a finite surjective morphism between integral Noetherian schemes, then the set $V\subseteq Y$ of points over which $f$ is flat is open. I want to show that this fails if we drop the finiteness assumption. I can think of an example given by blowing up $\mathbb A^3$ at a...