Let $X$ be a compact complex manifold, and with the following sheaves $\mathscr O$, the sheaf of holomorphic function, $\mathscr O^*$, the sheaf of nonvanishing holomorphic function $\mathscr K^*$ the sheaf of nonidentically zero meromorphic function. A divisor is an element in $\Gamma(X, \mathscr K^*/\mathscr O^*)$ (Cartier divisor). The short exact sequence of sheaves $$ […]

Let $X$ be an algebraic variety over an algebraically closed field $K$. By definition, $X$ is a separated prevariety, and $x \in X$. I’m trying to show (i): The minimal primes of $\mathcal O_{X,x}$ are in bijection with the irreducible components of $X$ containing $x$. Since $\mathcal O_{X,x}$ is reduced, this shows in particular that […]

Let $X$ be an affine algebraic set and $f\in K[X]$ where $K[X]$ is the coordinate ring of $X$. Suppose $I(X)=\langle G_1,\ldots,G_r\rangle$ and $W=Z(G_1,\ldots,G_r,FT_{n+1}-1)$, where $G_1,\ldots,G_r\in k[T_1,\ldots,T_n]$. The image of $F\in k[T_1,\ldots,T_n]$ in the quotient $K[X]$ is $f$, i.e, $f=F+I(X)$. I’m trying to prove these isomorphisms: $$k[W]\cong \frac{k[T_1,\ldots,T_n,T_{n+1}]}{\langle G_1,\ldots,G_r,FT_{n+1}-1\rangle}\cong \frac{k[X][T_{n+1}]}{\langle fT_{n+1}-1\rangle}\cong k[X][1/f].$$ I couldn’t prove it […]

The following exercise is taken from ravi vakil’s notes on algebraic geometry. Suppose $X$ is a closed subset of $\mathbb{P}^n_k$ of dimension at least $1$, and $H$ is a nonempty hypersurface in $\mathbb{P}^n_k$. Show that $H\cap X \ne \emptyset$. The clue suggests to consider the cone over $X$. I’m stuck on this and I realized […]

The relative chinese remainder theorem says that for any ring $R$ with two ideals $I,J$ we have an iso $R/(I\cap J)\cong R/I\times_{R/(I+J)}R/J$. Let’s take $R=\Bbbk [x_1,\dots ,x_n]$ for $\Bbbk $ algebraically closed. If $I,J$ are radical, the standard dictionary tells us $R(I\cap J)$ is the coordinate ring of the variety $\mathbf V(I)\cup \mathbf V(J)$. Furthermore, […]

This is the object that I am trying to find the volume of. This can be treated as a “26 point star”. What I need is an equation to describe it. If anyone has that surface in spherical coordinates R(Phi,Theta) that would be awesome. If not, the X,Y,Z solution can also be worked with. This […]

I am studying ideals such as toric ideals but I am unable to find a consistent definition, it seems to be very general so please explain the origin of “toric ideal”. Is there a geometric interpretation for toric ideal for example in three dimensions? Has toric ideal something to do with torus?

How to define the union of closed subschemes in an affine scheme? Suppose $I$ and $J$ define closed subschemes of $\operatorname{Spec}R$, how should we define their intersection? Eisenbud and Harris (GTM 197, p24) defined it by $I\cap J$ and used it to derive the “double points”(p60). We can also define it by $IJ$, which has […]

I am supposed to describe the points and topology of Spec$(\mathbb{R}[x])$, I managed to describe the points but I dont understand the “topology” of the set, what does this mean? Are they asking for the Zariski topology and how can I describe this?

Suppose $H_0$ is the hypersurface defined by a homogeneous polynomial $H$ in $\mathbb{P}^n(k)$. How do we show its complement $\mathbb{P}^n(k)_H$ is affine? (It is a problem in Mumford’s Redbook Ch1.5)

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