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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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When is a polynomial map proper?

Let $f\in \mathbb C[x_1,\dots,x_d]$ be a (nonconstant) polynomial. Of course it can be viewed as a (surjective) regular map $$\tilde f:\mathbb A^d_\mathbb C\to \mathbb A^1_\mathbb C.$$ Question. When is $\tilde f$ proper? The map $\tilde f$ being…
Brenin
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Zariski cohomology of $\mathbb{A}^1$ over a local ring with values in $\mathbb{G}_m$

Let $X$ be a the spectrum of a regular local ring. What is known about the vanishing of the Zariski cohomology group $$ H^n(\mathbb{A}^k_X,\mathbb{G}_m) $$ for $n,k\geq 0$? If $X$ has dimension $d$ and if $n>k+d$ so that $\mathbb{A}^k_X$ has…
user8463524
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Geometrical description of maps of schemes

In preparation for an exam, I am trying to solve the following question: Describe geometrically all maps from $\operatorname{Spec}(\mathbb{C}[z]/(z^2))$ to $\operatorname{Spec}(\mathbb{C}[x,y])$. I've been able to characterize all such maps: they…
Daniel Robert-Nicoud
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Hartshorne Ex III5.7 c)

Suppose $X$ is a reduced proper schemes over a noetherian ring, $X_i$ are its irreducible components，$L$ is a invertible sheaf on $X$. If $L|_{X_i}$ are all ample, how do we show $L$ is ample? If $X=X_1\cup X_2$, $F$ be a coherent sheaf on $X$, then…
user93417
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Thomason resolution of sheaves

Let $X$ be a smooth quasi-projective scheme over a field $k$ and $G$ an algebraic group (also over $k$ not necessarily reductive) acting on $X$. I the work of Thomason "Equivariant Resolution, Linearization, and Hilbert’s Fourteenth Problem over…
user109227
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Calculating a dual variety to a Chow variety

I am unfamiliar with algebraic geometry, yet I am faced with calculating three special cases of the following (the full text can be found at http://arxiv.org/abs/1107.4659) The $n^{\times d}$-hyperdeterminant of a symmetric tensor of degree $d \ge…
Paul Sundheim
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blowing-up and tangent cone: essentially identical concepts?

After reading about the concepts of blowing-up and tangent-cone of a curve at a point $P$, i have the following understanding: The blowing-up gives us the slopes of the tangent(s) of the curve at point $P$, while the tangent cone gives us the…
Manos
  • 25,833
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Intersections on General Nonsingular Projective Varieties

Let $X$ be a nonsingular, integral projective variety of dimension at least 2 over $k$ algebraically closed. Let $Y$ and $Z$ be two codimension 1 subschemes (effective Weil divisors) of $X$. Must they intersect? Moreover, letting $Y$ and $Z$ be…
Cass
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Hartshorne Example I.5.6.3

This question is concerned with Example I.5.6.3 in Hartshorne. Let $g, h$ be elements of $k[[x,y]]$ of the form $g = y+x+g_2+g_3+\cdots, h = y-x + h_2+h_3+\cdots$ where $g_i,h_i$ are homogeneous polynomials of degree $i$. Hartshorne writes "Since…
Manos
  • 25,833
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Is the base extension to K of an irreducible nonsingular projective variety over k irreducible?

Suppose $X$ is an irreducible nonsingular projective variety over a field $k$ (not necessarily algebraically closed) Let $K$ be a field extension of $k$ ( If $K/k$ is not algebraic, we can assume that the field $k$ is infinite. I don't know if this…
Suhas
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Unique line through two points in projective space

I'm trying to solve exercise I.3.15 in Hartshorne's Algebraic Geometry. The question starts as follows: Projection from a point: Let $ \mathbb{P}^{n } $ be a hyperplane in $ \mathbb{P}^{n+1 } $ and let $ P \in \mathbb{P}^{n +1 } - \mathbb{P}^{n }$.…
PeterM
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Exceptional locus of birational morphism is a divisor.

Let $f: V\to W$ be a proper birational morphism of smooth varieties, in a paper I'm reading the author claims that the exceptional locus of $f$ (i.e. the inverse image of the smallest closed set of $W$ outside of which $f$ is an isomorphism) is an…
ADR
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Is $c = \dim(X)$ if $P \otimes \mathcal{L} = P[c]$

Let $P \in \mathcal{D}^b(X)$ be an element of the derived category of coherent sheaves on a smooth projective variety. Suppose there is a line bundle $\mathcal{L}$ such that $P \otimes \mathcal{L}[\dim(X)] \cong P[c]$ for an integer $c$. Does it…
user5262
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2 answers

free module implies surjective map of affine schemes

If $A\to B$ is such that $B$ is a free $A$-module, is it true that $Spec(B)\to Spec(A)$ is surjective? I suspect it is true that there is a projection $B\to A$ so that the composition $A\to B\to A$ is the identity map (of rings), but I'm not sure.
LCL
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on the coordinate ring of $\mathbb{A}^n \times \mathbb{P}^{m}$

Consider the product $\mathbb{A}^n \times \mathbb{P}^{m}$. Let $x_i$ be affine coordinates on $\mathbb{A}^n$ and $y_j$ homogeneous coordinates on $\mathbb{P}^{m}$. Question: Is $A=k[x_1,\dots,x_n,y_0,\dots,y_m]$ the coordinate ring of…
Manos
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