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Questions tagged [algebraic-geometry]

The study of geometric objects defined by polynomial equations, as well as their generalizations: algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc. Problems under this tag typically involve techniques of abstract algebra or complex-analytic methods. This tag should not be used for elementary problems which involve both algebra and geometry.

The study of geometric objects defined by polynomial equations, as well as their generalizations: algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc. Problems under this tag typically involve techniques of abstract algebra or complex-analytic methods. This tag should not be used for elementary problems which involve both algebra and geometry.

29074 questions
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Is EGA II. 5.6.1's Statement of Chow's Lemma Correct?

It states that if $X\rightarrow S$ is a separated morphism of finite type, with $S$ quasi-compact and $X$ having finitely many irreducible components, then there is a surjective and projective $S$-morphism $f: X'\rightarrow X$, such that $X'$ is…
D D
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Locally closed embedding whose image is closed is a closed embedding?

Let $\pi: X \to Y$ be a locally closed embedding of schemes. Then we can realize $X$ as a closed subscheme of an open subscheme $U$ of $Y$, and $\pi$ factors as $$X \to U \to Y$$ where $X \to U$ is a closed embedding and $U \to Y$ is an open…
user5826
  • 11,982
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Degree and dimension of intersection of projective variety and hypersurface

I am looking at Theorem 7.7 of Hartshorne where he states the general form of Bezout's Theorem. The hypotheses of the theorem are as follows. Let $H$ be a hypersurface of degree $d$ and $Y \subseteq \Bbb{P}^n$ a projective variety of dimension $r$.…
user38268
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1 answer

$t$ - th graded piece of the coordinate ring of $Y \times Z$

Let $Y \subseteq \Bbb{P}^n$ and $Z \subseteq \Bbb{P}^m$ be two projective varieties. By $Y \times Z$, we really mean the image of $Y \times Z$ via the Segre embedding $\psi$ in $\Bbb{P}^N$ with $N = (n+1)(m+1) - 1$. We want to determine the $t$ -th…
user38268
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1 answer

Theorem on the Dimension of Fibers for locally closed an open sets

I map a locally closed set $X$ on an open set $Y$ by a polynomial map $f$. I know that for $y\in Y$, $\dim f^{-1}(y)=0$ (finite number of points). Is it true that $\dim X = \dim Y$? I found the Theorem on the Dimension of Fibers in Shafarevich's…
Ignat Domanov
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1 answer

Exercise 9.3.H in Ravi Vakil’s Foundations of Algebraic Geometry.

I am following the hint given in Exercise 9.3.H of Ravi Vakil’s notes. It can be found on page 261, here. The exercise states: any finitely presented morphism $\pi:X\to\operatorname{Spec} B$ is a pullback of a finite type morphism…
Fiona
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when is the ideal $I(\overline{V})$ of the projectivization of an affine variety generated by the homogenization of the generator of $I(V)$

Let $k$ be a field and $V \subset \mathbb{A}^n(k)$ an affine variety defined by polynomials $f_1,\cdots, f_k$. Under which conditions the ideal $I(\overline{V})$ of the projective completion $\overline{V}\subset \mathbb{P}^n(k)$ of $V$ is…
mika
  • 41
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Formal completion as a functor

Let $X$ be a scheme with closed subscheme $Z$. There is a natural way to think of $X$ as a functor from schemes to sets, $$X : S \mapsto X(S) = \mathrm{Mor}(S,X).$$ It seems there will be a similar way to understand the completion $\hat X$ of $X$ at…
user82625
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Counter example of upper semicontinuity of fiber dimension in classical algebraic geometry

We know that if $f : X\to Y$ is a morphism between two irreducible affine varieties over an algebraically closed field $k$, then the function that assigns to each point of $X$ the dimension of the fiber it belongs to is upper semicontinuous on…
brunoh
  • 2,436
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Why this object is a sheaf?

I whould like to know why $ \ \mathcal{C} : U \to \mathcal{C} ( U , \mathbb{R} ) $ is a sheaf ? $ U $ is an open set of $ E $ a $ \mathbb{R} $ - vector space which has a finite dimension. $ \mathcal{C} ( U , \mathbb{R} ) $ contains continous maps…
Bryan
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Does "low dimension" of a constructible set of $\mathbb C^n$ imply "zero measure" of the set in $\mathbb C^n$ and $\mathbb R^n$?.

I know that if some specific statement $S$ holds for $x\in\mathbb C^n$, then $x$ belongs to a constructible set $X\subset C^n$ (in Zariski topology) of dimension strictly lees than $n$. ${\bf Question\ 1:}$ Can I say that the statement $S$ does…
Ignat Domanov
  • 1,123
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The projection formula in Algebraic Geometry

In this book, the projection formula stated as follows; Let $f:X\to Y$ a separated, quasi-compact morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on X, $\mathcal{G}$ be a quasi-coherent sheaf on $Y$. Then…
User0829
  • 1,359
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Is "locally of finite type" affine-local on the source?

Hartshorne exercise II.3.3(c) asks the reader to prove that if $f:X\to Y$ is finite type, then for any $\mathrm{Spec}A\subset Y$ and $\mathrm{Spec}B\subset X$ with $\mathrm{Spec}B\subset f^{-1} \mathrm{Spec}A$, we have $A\to B$ is of finite…
Xander Flood
  • 1,125
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Serre's criterion for affinity

I am studying Algebraic Geometry with this book. Through the Serre's criterion for affinity, we can know that $H^1(X,\mathcal{F})=0$ then $X$ is affine with some conditions on $X$ and $\mathcal{F}$. I have a question about the proof. In the book,…
User0829
  • 1,359
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1 answer

Exceptional Divisor of Node Blow-Up is Two Smooth Reduced Points

This is Vakil 29.3 D, self-study. We are to show that if $X$ is a variety over an algebraically closed field $k$ with a node at point $p$, that the blow-up of $X$ at $p$ gives a morphism $$\beta: \tilde{X} \to X$$ such that the exceptional divisor…
Johnny Apple
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