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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 an affine bundle trivial?

Let $k$ be a field, not necessarily algebraically closed, not necessarily of characteristic 0 (actually, the example I have in mind is $k=F_2$). Let $V,W$ be varieties over $k$, and $W\to V$ a morphism defined over $k$, such that every fiber is…
Charles Siegel
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Cohomology with support and Poincare duality

Say we have hypersurface $D \subset \mathbb{P}^2$. I ran into the following sequence of isomorphisms, justified only with "by Poincare duality formulated algebro-geometrically": $$ H^2(\mathbb{P}^2-D,\mathbb{Q}) \cong H^3_D(\mathbb{P}^2,\mathbb{Q})…
baltazar
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Why does the virtual fundamental class deserve its name?

I'm reading about virtual fundamental classes, following Behrend and Fantechi's approach in their paper, The Intrinsic Normal Cone. I understand that these classes enjoy the following desirable properties They refine the top chern class (related…
user041193
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an irreducible quadric $ X \subset \Bbb A^n$ d is birational to some $\Bbb A^m$

I want to prove that an irreducible quadric $ X \subset \Bbb A^n$ defined by a quadratic equation $ F(T_1,\ldots,T_n)=0$ is rational (i.e birational to some affine space $\Bbb A^m$ ). I'm not sure how to do this, the book "Algebraic Geometry of…
Daniel
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What is the least number of symmetric polynomials needed to determine a unique solution z_1,...,z_n?

Is there a system $\{s_1, \cdots, s_m\}$ of symmetric polynomials of $z_1, \cdots, z_n \in \mathbb{C}$ such that $$s_1(z_1, \cdots, z_n) = c_1$$ $$s_2(z_1, \cdots, z_n) = c_2$$ $$\cdots$$ $$s_m(z_1, \cdots, z_n) = c_m$$ has at most one solution…
charles.y.zheng
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Reference request for a proof of a basic version of normalization lemma

I was reading some notes on affine algebraic geometry and came across the following fact: If $k$ is a field with $\mathrm{char}~k = 0$, then if $V(f)$ is an irreducible hypersurface in $\mathbf{A}^n_k$, then there is a surjective linear projection…
Drew Brady
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divisors and powers of line bundles

Can anyone help me with the following question? Let $X$ be a smooth, projective algebraic variety. Let $D$ be an effective divisor on $X$ and $m$ an integer. Under which conditions there exists a line bundle $L$ such that $\mathcal{O}_X(D)=L^m$?…
cruella
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Set theoretic image of the structure morphism of a fiber product of schemes

Let $S$ be a scheme. Let $Z = X\times_S Y$ be the product of $S$-schemes. Let $f \colon X \rightarrow S, g\colon Y \rightarrow S, h\colon Z \rightarrow S$ be the structure morphisms. Then $h(Z) = f(X) \cap g(Y)$?
Makoto Kato
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Equivalent Statements for a Coherent Sheaf

Let $C$ be a curve and $f:C \to \mathbb{P}^1$ a finite morphism. I know that the induced sheaf $\mathcal{A}= f_*(\mathcal{O}_C)$ is coherent. I want to know why following statements are equivalent: 1) $\mathcal{A}= f_*(\mathcal{O}_C)$ is locally…
user267839
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Second Zariski cohomology of the multiplicative group

Let $X$ be a scheme, then $H^2_{et}(X, \Bbb G_m)$ is defined as the Brauer group of $X$ and people always say that Zariski topology is so coarse that we need to consider etale cohomology in practice. However, we have $H^1_{et}(X, \Bbb G_m) \cong…
user395911
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Exercise II-13 of Eisenbud-Harris Geometry of Schemes

I'm a topologist who is learning a bit of algebraic geometry for fun (to pass the time during my Christmas break), and I'm stuck on Exercise II-13 of Eisenbud-Harris's "The Geometry of Schemes". Let $\mathcal{H}$ be the set of finite subschemes of…
Adam Smith
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Confusion of the definition of a completely regular polygon $P$

I am a student studying an algebra & geometry module with an exam at the end of January. I have noticed that there are two alternate definitions of a completely regular polygon $P$: A topological space $X$ such that for every closed subset $C$ of…
user515496
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1 answer

Question on projective closure

Let $Y\subseteq \mathbb{A}^n$ be an affine variety, $\bar{Y}$ be its projective closure in $\mathbb{P}^n$. In GTM 52 of Robin Hartshone, there is a problem of finding generator of $I(\bar{Y})$(problem 2.9) Suppose that $\varphi : \mathbb{A}^n…
Arsenaler
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Hyperplane doesn't contain component of a scheme

Suppose that $I$ is ideal sheaf of $\mathcal{O}_\Bbb{P^r}$. Let $X$ be a scheme defined by $I$. Is it true that we can find hyperplane $H$ such that it doesn't contain any components of $X$?
david
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Number of prime divisors of a proper closed subset of a scheme

Let $X$ be a noetherian integral separated scheme which is regular in codimension one. Let $Z$ be a proper closed subset of $X$. There is a claim I could not figure out: $Z$ can contain at most finitely many prime divisors of $X$ (as usual, a prime…
User0829
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