Mathematics Stack Exchange
Mathematics Stack Exchange

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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Induced scheme structure on an irreducible component?

Suppose that $X$ is a non-reduced scheme of finite type over a field, with multiple irreducible components $X_1,\ldots,X_n$, possibly intersecting each other. Is there a natural scheme structure on each $X_i$? I don't want the reduced induced…
tim p
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Example of a non-normal integral variety $X$ for which "algebraic Hartog's" theorem fails?

In particular, I am interested in seeing an example of a rational function defined on a integral variety $X$ that is regular on the complement of a codimension $2$ closed set, but so that there is no extension to a regular function defined on all of…
Elle Najt
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How to understand push-forward an exceptional divisor?

Let X be a Gorenstein variety. And $f:Y\longrightarrow X$ is a resolution, Y is normal. And we have $K_Y=f^*K_X+\sum_{i}a_iE_i$, $E_i$ are exceptional divisors. Then consider the push-forward of the line bundle $K_Y$ (since Y is Gorenstein, K_Y is…
Shuhang
  • 664
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Restricted sheaf on a subset

Let $F$ be a sheaf of $k$-algebras on a topological space $X$. So by definition, $F$ is a contravariant functor from the category of open sets of $X$ to the category of $k$-algebras, satisfying a certain patching condition and a uniqueness…
D_S
  • 33,891
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Questions about elliptic fibrations

Say I have an algebraic surface $S$ (smooth, projective over $\mathbb{C}$) and a morphism $f: S \to B$ to some curve $B$ (smooth, projective over $\mathbb{C}$) defined by the complete linear system corresponding to some divisor $D$ on $S$. We assume…
Evariste
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Tangent space of a projective variety is well-defined

For $X$ an affine variety and $p \in X$ define $T_p X = \mathrm{Der}(k[X], \mathrm{ev}_p)$. Claim: If $Y = X \setminus Z(f)$ is some Zariski open affine subvariety of $X$ and $p \in Y$, then $T_p X \cong T_pY$. Definition: Let $X$ be an arbitrary…
Matt
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Example of a normal variety (or scheme) which is not locally factorial?

I am going to assume that in any case all schemes are Noetherian, separated and integral. Can someone provide an example of a scheme that is normal but not locally factorial. I know that being locally factorial will imply normal, because UFDs are…
Elle Najt
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Why is the secant variety a variety?

As I understand, the first secant variety of a variety $\mathbb{V}$ is constructed in the following way: Take two arbitrary points on the variety, construct the line passing through them, the closure of all such lines is then the first secant…
kaiser
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Composition of flat morphisms

Suppose I have nice enough schemes $X,Y,Z$ over a field $k$ (I can take $X,Y,Z$ to be smooth projective connected varieties and $k$ to be algebraically closed of characteristic 0 if needed) and morphisms of $k$-schemes $$X \xrightarrow{a} Y…
lostmathematician
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Intersection numbers on a surface

Some (probably very easy) questions on intersection theory on surfaces... Say $S$ is a smooth projective surface over $\mathbb{C}$ with canonical divisor $K_S$. If $S$ is not ruled and $H$ is a hyperplane section (for an arbitrary embedding), do we…
Evariste
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Definition of smooth morphism; geometric fibers and separatedness

Wikipedia (https://en.m.wikipedia.org/wiki/Smooth_morphism) defines a smooth morphism of schemes as one that is locally finitely presented (lft) and flat with regular (=smooth) geometric fibers. But the article goes on to say that this is equivalent…
LCL
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If $X_1, \ldots, X_n$ are a collection of codimension $\geq 2$ subvarieties of $P^n$, is there an irreducible variety containing them?

If $X_1, \ldots, X_n$ are a collection of codimension $\geq 2$ subvarieties of $P^n$, is there an irreducible hypersurface containing them? I would be satisfied with an answer to : if $x_1, \ldots, x_n$ are points in the plane $P^2$, is there an…
Elle Najt
  • 20,740
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What does Mumford mean by "an extension" here?

From Mumford's Red Book, Chapter 2, Example K: Take $X = Y = \mathbb{P}^2$, and let $x_0, x_1, x_2$ and $y_0, y_1, y_2$ be homogeneous coordinates on $X$ and $Y$. Let $U_0 \subset X$ and $V_0 \subset Y$ be defined as the open sets $x_0 x_1…
Daniel McLaury
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Dimension of quotient in projective space

Let $k$ be an algebraically closed field with characteristic distinct from $2$. I want to compute the multiplicity of the intersection of the points which lie in $V_{1} \cap V_{2}$ where: $V_{1}=V(x^{2}+y^{2}-z^{2}) \subseteq \mathbb{P}^{2}$ and…
user31509
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fiber products of curves

Let $X,Y,Z$ be three nonsingular curves over a field $k$ (not necessarily proper, ie, possibly affine). Let $f : X\rightarrow Z$ and $g : Y\rightarrow Z$ be finite morphisms. We know the fiber product $X\times_k Y$ is a nonsingular surface. The…
oxeimon
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