# All questions with tag [math: algebraic-geometry]

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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 $f \in I(... 2022-07-25 20:47:17 0 ## How to find a finite set of generators for$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)$by Hilbert's nullstellens... 2022-07-25 20:47:06 0 ## An exercise involving the twisted cubic One of the exercises in Hatcher's Harris's AG book goes as follows. Let$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
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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 $dF_1|_p,\dots... 2022-07-25 20:46:22 0 ## Example of a Singular points and its locus A factor$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 1 ## 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 1 ## 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
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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 $... 2022-07-25 20:42:46 0 ## product ampleness let X a smooth relative projective scheme over a normal basis S. Let$\{(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 &gt;0$and that$r_1\geq r_2$. Is it true that$(L_1\otimes L_2)^d$is very ample ... 2022-07-25 20:42:42 0 ## Is Transversality invariant by losing Dimension? Setup Let$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
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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 $X \to \mathbb{P}^2... 2022-07-25 20:40:49 0 ## Sufficient condition for surjectivity of a morphism of group schemes Let$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$. I would like a reference for this fact. Remarks As suc... 2022-07-25 20:38:41 1 ## stereographic projection in a projective curve It appears that there is a typical strategy for a suggestion comparable to the stereographic estimate. I do not recognize just how can I utilize it. As an example below in this workout, just how can I utilize it? I'm actually sorry for being so foolish Define a birational map from an irreducible quadric hypersurface$X \subset P^3$to$P^2$, by... 2022-07-25 20:23:10 0 ## Why topological stratification is useful? My major emphasis gets on the applications of in complex/abstract Algebraic geometry specifically, from the system - logical point of view and also (Added) Moduli rooms. I have an obscure sensation that just how decaying our (topological) room will certainly serve specifically, for researching selfhoods, additionally considering the offered r... 2022-07-25 20:22:01 1 ## smooth K3 surface In his paper "", Jan Christian Rohdes asserts that the surface area$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$... 2022-07-25 20:20:37 0 ## example of morphism and varieties So I have two varieties;$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
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## 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
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## 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
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## 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
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## 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