All questions with tag [math: algebraic-geometry]


properties of ideals in $K[x_1,\ldots ,x_n]$

I'm trying to convince myself of the 4 following facts: If $X \subseteq Y \subseteq A^n_{k}$ then $I(Y) \subseteq I(X)$ If $J \subseteq K[x_1,\ldots,x_n]$ is an ideal then $J \subseteq I(V(J))$ If $X \subseteq A^n_{k}$ then $X \subseteq V(I(X))$ $X=V(I(X))$ if and only if $X$ is an algebraic set My attemps to prove them: 1) Let $f \in I(...
2022-07-25 20:47:17

How to find a finite set of generators for $I \subset k[x_1, ..., x_n]$

Suppose that we have a set of polynomials $f_1, ..., f_m \in C = k[x_1, ..., x_n]$ where $k$ is an algebraically closed field, and let $V$ denote the set on which they all simultaneously vanish. Define $I(V)$ to be the ideal of polynomials in $C$ which vanish on all of $V$. In general, $I(V) = \text{rad}(f_1, ..., f_m)$ by Hilbert's nullstellens...
2022-07-25 20:47:06

An exercise involving the twisted cubic

One of the exercises in Hatcher's Harris's AG book goes as follows. Let $F_0 = Z_0 Z_2 - Z_1^2$, $F_1 = Z_0 Z_3 - Z_1 Z_2$, $F_2 = Z_1 Z_3 - Z_2^2$ (s.t. $\mathbb{V}(F_0, F_1, F_2)$ is the twisted cubic). Define $F_\lambda = \lambda_0 F_0 + \lambda_1 F_1 + \lambda_2 F_2$. Prove that for any $[\lambda_0, \lambda_1, \lambda_2] \neq 0$ and $[\mu_0,...
2022-07-25 20:46:44

projective tangent space to $V$

I am not obtaining the trouble just how and also where to begin with, will certainly be pleased for your aid: $V$ is a proj.variety in $\mathbb{P}^n$ whose uniform extreme perfect is created by uniform polynomials $F_1,\dots, F_m$. Show that the projective tangent room to $V$ at $p$ is specified by the uniform straight polynomials $dF_1|_p,\dots...
2022-07-25 20:46:22

Example of a Singular points and its locus

A factor $p$ on a seemingly - projective selection $V$ is called smooth if $$dimT_p(V)=dim_p(V)$$ or else $p\in V$ is single. Could any kind of one clarify me with an instance of a single factor and also just how their locus creates a shut part of $V$?
2022-07-25 20:46:07

are fibres of a flat bijective map reduced?

Let $f: X \to Y$ be a level map of algebraic selections or of intricate analytic rooms which is bijective on shut factors (or simply bijective in the secnond instance). Intend both $X$ and also $Y$ are lowered. Is it real that $f$ has lowered fibers? If it holds true, I would certainly be most happy for a reference.
2022-07-25 20:45:52

Is $f$ reduced if and only if the derivations $\gcd(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y})=1$ under some conditions?

I have encountered the following problem. I have no ideas to prove it or disprove it. Suppose that $f\in \mathbb{C}[x,y]$, $f(0,0)=0$, $\frac{\partial f}{\partial x}(0,0)=0, \frac{\partial f}{\partial y}(0,0)=0$. Suppose $f$ is square-freeā€”for simplicity we may first assume that $f$ is irreducible. Then $\frac{\partial f}{\partial x}$ and $\...
2022-07-25 20:43:48

DVR-valued points of schemes

Let $X$ be a scheme of finite type over a discrete valuation ring $R$ with fraction field $K$, such that the generic fibre $X_K$ is smooth over $K$. Let $Y$ be a closed subscheme of $X$ which contains no irreducible component of $X$. Is it true - maybe under some additional assumptions on $X$ and/or $R$ - that $X(R) \setminus Y(R)$ is dense in $...
2022-07-25 20:42:46

product ampleness

let X a smooth relative projective scheme over a normal basis S. Let $\{(Y_i,L_i)\}_{i=1,2}$ be schemes with relatively projective morphisms $Y_i\rightarrow X$ induced by ample line bundles $L_i$ on $Y_i$. Assume that $L_i^{d_i}$ is very ample for $d_i\geq r_i >0$ and that $r_1\geq r_2$. Is it true that $(L_1\otimes L_2)^d$ is very ample ...
2022-07-25 20:42:42

Is Transversality invariant by losing Dimension?

Setup Let $X$ be a smooth, reduced and irreducible scheme. And let $E$ be a vector bundle of rank $n$ on $X$. Following Eisenbud and Harris we say that the collection $\{\sigma_1,...,\sigma_k\}$ of sections of $E$ is dimensionally transverse if $codim_X(V(\sigma_1\wedge ...\wedge \sigma_k))=n-k+1$. Here $V(\sigma_1\wedge ...\wedge \sigma_k):=\{...
2022-07-25 20:42:35

projection of a quadric surface

Consider the quadric surface $X = \{ xy = zw \} \subset \mathbb{P}^3$ and pick a point $x \in X$. I think it is true that if we think of $\mathbb{P}^2$ as the space of lines through $x$ in $\mathbb{P}^3$, then the morphism $X \setminus \{ x \} \to \mathbb{P}^2$ which sends $y \mapsto \overline{xy}$ represents a birational map $X \to \mathbb{P}^2...
2022-07-25 20:40:49

Sufficient condition for surjectivity of a morphism of group schemes

Let $G$ be a group scheme over a field $F$, and let $f:G\to G$ be a homomorphism. Written in my notes, I have the following statement: To check surjectivity (on $F$-rational points), it suffices to show that the induced morphism $G_K\to G_K$ is surjective, for some finite extension $K/F$. I would like a reference for this fact. Remarks As suc...
2022-07-25 20:38:41

stereographic projection in a projective curve

It appears that there is a typical strategy for a suggestion comparable to the stereographic estimate. I do not recognize just how can I utilize it. As an example below in this workout, just how can I utilize it? I'm actually sorry for being so foolish Define a birational map from an irreducible quadric hypersurface $X \subset P^3$ to $P^2$, by...
2022-07-25 20:23:10

Why topological stratification is useful?

My major emphasis gets on the applications of in complex/abstract Algebraic geometry specifically, from the system - logical point of view and also (Added) Moduli rooms. I have an obscure sensation that just how decaying our (topological) room will certainly serve specifically, for researching selfhoods, additionally considering the offered r...
2022-07-25 20:22:01

smooth K3 surface

In his paper "", Jan Christian Rohdes asserts that the surface area $S \subset \mathbb{P}^3$, with variables $(y_2: y_1: x_1: x_0)$, offered by the list below formula is smooth $(y_2^3-y_1^3)y_1+(x_1^3-x_0^3)x_0=0$ He claims: "By making use of partial by-products of the specifying formula, one can conveniently validate that $S$ ...
2022-07-25 20:20:37

example of morphism and varieties

So I have two varieties; $V=V(y-x^2)$ and $W=V(y^2-x^3)$ $\phi: V \mapsto W$; $(x,y) \mapsto (y,xy)$ define $\phi^*: C[W] \mapsto C[V]$; $[f] \mapsto [f o \phi]$ This is the morphism i'm looking at; $\gamma: V \mapsto W$ by $(x,y) \mapsto (x^2,xy)$ Can anyone tell me what $\gamma(x,y)$ is? Also, how do you compute $\gamma^*:C[W] \mapsto ...
2022-07-25 20:20:22

equivariant hyperplane sections

Suppose you are given a smooth algebraic variety $X$ inside a projective space $\mathbb{P}$ and that there is a linear action of a finite cyclic group $G$ on $\mathbb{P}$ which restricts to an action on $X$. Does there exists an hyperplane $H$ on $\mathbb{P}$ such that $H \cap X$ is smooth and stable under $G$? I guess it should be possible to p...
2022-07-25 20:18:09

Scheme flat of finite type over $\mathbb{Z}$

Let $X$ be a scheme which is integral, of finite type, flat and separated over $\mathbb{Z}$. Let $D \subseteq X$ be a prime divisor on $X$ which is not flat over $\mathbb{Z}$. Is it true that $D(\mathbb{F}_p) = \emptyset$ for all primes $p$, with at most one exception?
2022-07-25 20:16:04

projective cubic

I have some troubles to confirm that the photo of the function $f:\,\mathbb P^1\longrightarrow\mathbb P^3$ such that $$(u,v)\longmapsto (u^3,\,u^2v,\,uv^2,\,v^3)$$ is the algebraic projective set $$V(XT-YZ,\, Y^2-XZ,\,Z^2-YT)$$ Clearly I have trouble to confirm that the algebraic projective set is had in the photo of $f$. Specifically "solv...
2022-07-25 20:16:00

Proving well-definedness of "valuation of $f$ at $p$"?

Let $X$ be an irreducible variety, $p \in X$. Define $\mathcal{O}_{X,p}$ and $\mathcal{m}_{X,p}$ as usual. We have the following theorem: $\mathcal{m}_{X,p} = (\pi)$ is a principal ideal and $\bigcap _{n \geq 0 } \mathcal{m}_{X,p}^n = \{ 0 \}$. I'm trying to understand the proof of the following: Every $f \in k(X)^\times$ can be written uniquel...
2022-07-25 20:15:01