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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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Help with this proof in Fulton book

I'm really confused with the definitions of coordinate rings and field of rational functions. I'm trying to understand this proof which I was stuck in the very beginning: First I didn't understand the definition of $J_f$. we have $\overline…
user42912
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Hartshorne Exercise V.5.5 Every geometrically ruled surface over a fixed curve $C$ is birationally equivalent

Let $C$ be a curve, and let $\pi: X \rightarrow C$ and $\pi^{\prime}: X^{\prime} \rightarrow C$ be two geometrically ruled surfaces over $C$. Show that there is a finite sequence of elementary transformations (5.7.1) which transform $X$ into…
Shrugs
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Closed point and base change of scheme.

Let $k$ be a field and $K/k$ be a field extension. For a scheme $X$ of finite type over $k$, denote $X_K:=X\times_k \text{Spec}K$. Let $x\in X$ be a closed point and $x'\in X_K$ be a point lying over $x$. In this situation, is $x'$ also a closed…
User0829
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Blowing up of affine space

I am learning blowing up from An Invitation to Algebraic geometry (Karen E. Smith). In the chapter 7 (103- page) written that blowing up of affine space $\mathbb{A}^n$ along point $p$ is not affine variety. How I can prove it? I am trying to use the…
Otabek
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How to calculate the Picard group of the following variety

I am puzzled by the Picard group of $X=\text{Proj }k[x_0,x_1,x_2,x_3,x_4]/(x_0x_1-x_2x_3)$. I can calculate its Weil's divisor class group ($\mathbb{Z}\oplus \mathbb{Z}$) because it is the projective cone of $\text{Proj…
user884626
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Scheme theoretic image

Let $f : Z \rightarrow X$ be a morphism of schemes. I want to get a unique closed subsheme $Y$ of $X$ with the following propertie : the morphism $f$ factor through $Y$ and if $Y'$ is any other closed subscheme of $X$ through which $f$ factors, then…
A.G
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Underlying space of fiber product of schemes

Let $X,Y$ be schemes over a field $k$. I know that $X \times_k Y$ is not equal to the catesian product $X \times Y$. For example $\mathbb{A}_{k}^2 \neq \mathbb{A}_{k} \times \mathbb{A}_{k}$. So, I don't understand example II.4.0.1 in Hartshorne's…
Sang Cheol Lee
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When is the intersection of two irreducible variety irreducible?

I am learning AG. At first, I thought the intersection of two irreducible varieties was irreducible. This isn't true in general. For instance, take the parabola and a general line. They will intersect at two points, so it isn't irreducible. I am…
Chef-
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Hyperplane sections of surfaces in $\mathbb{P}^3$

Suppose that $X \subset \mathbb{P}^3$ is a smooth complex projective surface of degree $d \geq 4$. Then the Noether-Lefschetz theorem states that if $X$ is very general, then $\mathrm{Pic}(\mathbb{P}^3) \rightarrow \mathrm{Pic}(X)$ is an…
Irwin
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Does a morphism that is constant on fibers factor?

Given a morphism $\varphi:Z\to Y$ over $X$, where $X,Y,Z$ are all varieties over an algebraically closed field of characteristic zero. Let $q:Z\to X$ be the structure morphism. Suppose that $q$ is smooth and $\varphi$ is constant on the fibers of…
schuming
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Is any $K3$ surface of degree $8$ in $\mathbb{P}^5$ the complete intersection of quadrics?

Here the base field is the complex numbers $\mathbb{C}$.
user8403
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Example of a restriction of sheaf not being a sheaf

Let $\mathscr{F}$ be a sheaf on $X$, and $Y\subset X$ a subset. Define a presheaf $\mathscr{F}|_Y$ on $Y$ via the direct limit $$\mathscr{F}|_Y(V):=\lim_{V\subset U}\mathscr{F}(U),$$ where $V$ is an open subset of $Y$, and $U$ is an open subset of…
ashpool
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Seeming gap in Hartshorne's Algebraic Geometry, proof of Proposition 1.10

The claim is that if $Y$ is a quasi-affine variety (an open subset of an affine variety), then dim $Y = $ dim $\overline{Y}$. Here dimension is defined as the maximal length of an ascending chain of irreducible closed subsets, minus one. Hartshorne…
Vik78
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Dimension of range of an function

Let $f$ be a rational function from affine variety $X$ to affine variety $Y$. Is it always true that $\dim X \geq \dim f(X)$? If it is can someone provide me with a proof of it? To me, this is intuitively true. Also, is this true for functions in…
user44322
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Example of a non-affine irreducible scheme

What are basic examples for irreducible schemes which are not affine? What happens if I also demand the scheme to be Noetherian and/or locally Noetherian?
Guest
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