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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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CounterExamples for Hilbert Nullstellensatz

It is easy to prove that for any two algebraic sets $X_1, X_2$ in $\mathbb{A}^n$ we have that $$I(X_1\cap X_2) = \sqrt{I(X_1)+I(X_2)}$$ Find an example that the radical is neccessary, i.e., an example on algebraic sets $X_1, X_2$ that $I(X_1\cap…
9999
  • 959
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What are prime ideals in $k[[t]]$ and $k((t))$?

Let $k$ be a field. What are prime ideals in $k[[t]]$ and $k((t))$? I think that $(t-a), a \in k$ and irreducible polynomials in $k[[t]]$ are prime ideals of $k[[t]]$. Are there other prime ideals? Thank you very much.
LJR
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Showing that the interesection of a linear subspace with a subvariety is non-empty

I am trying to prove 5.15b) here: http://www.math.u-bordeaux1.fr/~qliu/Book/Errata3/pages76-77.pdf I am not sure if my argument is complete, and would therefore appreciate if someone could look into it. Update: If you know the solution, but don't…
user101036
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Decomposing an affine algebraic set into irreducible ones

Let $X\subset\mathbb C^4$ be given by the system \begin{align} x_1x_4 - x_2x_3 &=0\\ x_1x_3 - x_2^2 &=0 \end{align} I need to decompose this into a union of irreducible sets. The obvious approach is to try and find the set by obtaining relations on…
Artem
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Some question of codimension 1

(1) "For affine variety $V$ of $\mathbb{A}^{n}$ such that its coordinate ring is UFD, closed subvariety of $V$ which has codimension 1 is cut out by a single equation." I looked at the proof of this statement, Where I have been using UFD in I…
Sang Cheol Lee
  • 3,119
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regular functions on a subvariety (problem 1.3.13 Hartshorne)

There is a post about this exact question here I am having the exact same issue and don't find the solution there complete. Let $Y\subseteq X$ be a subvariety. Let $\mathcal{O}_{Y,X}$ be the set of equivalence classes $\langle U,f\rangle$, where…
TheNumber23
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How to show that $\operatorname{Spec}(A)=\bigcup_{i=1}^n D(g_i)$ implies that $(g_1, \ldots, g_n)=A$?

$\newcommand{\Spec}{\operatorname{Spec}}$ Let $A$ be an algebra and $\Spec(A)$ the scheme consisting of all prime ideals of $A$. How to show that $\Spec(A)=\bigcup_{i=1}^n D(g_i)$ implies that $(g_1, \ldots, g_n)=A$? By definition, $D(g_i)=\{p \in…
LJR
  • 14,520
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How to find the ring of regular function on $\mathbb{P}^2\backslash\mathbb{V}(x_0^2+x_1^2+x_2^2)$

Let $X=\mathbb{P}^2$ and $U=X\backslash\mathbb{V}(x_0^2+x_1^2+x_2^2)$, could anyone show me how to find $\mathcal{O}_X(U)$? I see examples in affine case, but have no idea how to calculate the ring in the projective case.
hxhxhx88
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Hilbert polynomial, degree

my question concerns the Hilbertpolynomial of a coherent sheaf $F$ on a smooth projective variety $X$ with canonical bundle $\omega_X$: does the Hilbertpolynomial always have degree equal to $dim supp(F)$ or does in general only an inequality…
Descartes
  • 650
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Does a map between Jacobians come from a map of curves?

Let $Y_i=J(C_i)=\textrm{Pic}^0(C_i)$ be Jacobians of two smooth projective curves $C_1,C_2$. Suppose there is a morphism $Y_1\to Y_2$. Does it come from a morphism $C_1\to C_2$? I ask this because I have a curve $D$ mapping to a Jacobian $Y=J(C)$…
Brenin
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A question in Mumford Redbook P91

Suppose $K_0$ is a field isomorphic to the function field of prevariety$X,Y$. Choose $k-$isomorphism $\alpha:K(X)\to K_0; \beta: K(Y)\to K_0$, we get $A: \operatorname{Spec} K_0\to X, B:\operatorname {Spec}K_0 \to Y$, so we get $(A,B):…
user93417
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2 answers

Radical ideals and $\operatorname{Proj}$

It is well-known that in $\operatorname{Spec}A$, $V(I)\subset V(J)$ implies $\sqrt{I}\supset\sqrt{J}$. Is it also true in $\operatorname{Proj}A$, where $A$ is an $\mathbb{N}$-graded ring and $I,J\subset A$ are graded ideals? The difficulty in…
ashpool
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Surjectivity of a map

Let $C$ be a curve and $D\in\mathrm{Div}(C)$. Suppose $W$ is the canonical divisor. Let $g\in L(D)$ be such that $g\not\in L(D-P)$ for all $P\in C$ and $P\leq W-D$. Then one can show (given in Algebraic Curves by Fulton) that the following map is…
pritam
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Is complex Abelian variety isogenic to its dual?

Suppose $A$ is a complex Abelian variety and let $A^V$ be the dual Abelian variety. $A^V$ is defined as the moduli space of all line bundles on $A$ with $c_1=0$. It seems to me that $A$ is isogenic to $A^{V}$. But I am not sure that this is correct.…
agleaner
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Is the restriction map of structure sheaf on an irreducible scheme injective?

Suppose $X$ is an irreducible scheme, $U \subset V$ open subsets of $X$, does it hold that $\rho_U^V:O(V)\to O(U)$ injective? Generally under what conditions does it hold? Actually it is related to an exercise in Liu Qing's book p67,Ex4.11: Let…
user93417
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