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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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Why is first cohomology group of divisor sheaf on riemann surface zero?

Let $X$ be riemann surface (not supposed compact) and $\mathcal D$ be sheaf of divisors on $X$.Remind that this means for $U\subset X$ open then $\mathcal D(U)$ is group of divisors on $U$. How to prove that $H^1(X,\mathcal D)=0$ . This is exercise…
evgeniamerkulova
  • 2,062
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Noether Normalization and rational points over finite fields

Let $k$ be a finite field with $q$ elements, $U$ an integral scheme of finite type over $k$ and $f: U \to \mathbb{A}_k^d$ a finite dominant morphism (obtained by Noether normalization, $d=\dim U$). If $K$ is an extension of $k$ of degree $n$, how…
user1728
  • 451
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Separatedness of a composition, where one morphism is surjective and universally closed.

I'm stuck with the following problem: Let $f:X \rightarrow Y$ and $g:Y \rightarrow Z$ be scheme morphisms such that f is surjective and universally closed and such that $g \circ f$ is separated. The claim is then that g is also separated. I've been…
Dedalus
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Pushforward and pullback of invertible sheaf on ruled surface

Suppose $\pi:X\to C$ is a geometrically ruled surface, and $D$ a divisor on $X$. Then if $D.f=0$ for a fibre $f$, we know by Grauert's theorem that $\pi_{*}(\mathscr{L}(D))$ is a invertible sheaf on $C$. It is know that…
Xuqiang QIN
  • 968
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Finding $a^5 + b^5 + c^5$

Suppose we have numbers $a,b,c$ which satisfy the equations $$a+b+c=3,$$ $$a^2+b^2+c^2=5,$$ $$a^3+b^3+c^3=7.$$ How can I find $a^5 + b^5 + c^5$? I assumed we are working in $\Bbb{C}[a,b,c]$. I found a reduced Gröbner basis $G$: $$G = \langle…
Patrick Shambayati
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How do we describe the point 1 on the affine line?

The affine line $\mathbb{A}^1$ (over $Spec R$) is, as far as I understand, $Spec R[x]$. My guess is that $0\in \mathbb{A}^1$ is given by the (opposite of the) composite homomorphism $R[x] \to R \to F$ where the first map sends a polynomial to its…
theHigherGeometer
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The projective space is not affine (II)

This question is closely related to Projective space is not affine. I want to show that the projective space is not affine and to this end I want to prove that $\Gamma(\mathbb P^n_R, \mathcal O_{P_R^n})=R$. Intuitively, I get that the only…
user306194
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Prove that all automorphisms of the line $\Bbb A^1$ are of the form $f(x) = ax + b$ with $a\neq 0$.

An isomorphism $f : X → X$ of a closed set $X$ to itself is called an automorphism. Prove that all automorphisms of the line $\Bbb A^1$ are of the form $f(x) = ax + b$ with $a\neq 0$. I think I can take the co-ordinate ring and again closed subsets…
Ri-Li
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Is an elliptic curve canonically isomorphic to its quotient by its own $n$-torsion?

Let $E$ be an elliptic curve (say, over a field $K$), and $E[n]$ be its $n$-torsion subgroup-scheme (suppose char $K$ is coprime to $n$). Is $E$ canonically isomorphic to $E/E[n]$? (What is this canonical isomorphism?) I feel like there should be a…
oxeimon
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Multiplicity under polynomial maps

I'm reading Fulton's book and I'm trying to solve the following exercise : Let $T : \Bbb A^2 \rightarrow \Bbb A^2$ be a polynomial map, $T (Q) = P$. (a) Show that $m_Q(F^T ) \geq m_P (F)$. (b) Let $T = (T1,T2)$, and define $J_QT = (\partial T_i…
Ghassen Hamrouni
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Is Proj of the tensor product of two graded $A$-algebras isomorphic to the fiber product?

Some sanity checks: If $S_*$ and $T_*$ are two $A$-algebras, then is $Proj(S_* \otimes_A T_*) \cong Proj(S_*) \otimes_A Proj(T_*)$? Here the tensor product means $(S_* \otimes T_*) = \oplus_{n \geq 0} \oplus_{i + j = k} S_i \otimes_A S_j$. (I know…
Elle Najt
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Closed points of $Spec(A)$ are dense

This is exercise 3.6.J from the most recent Vakil's notes. Suppose that k is a field, and A is a finitely generated k-algebra. Show that closed points of Spec A are dense, by showing that if f ∈ A, and D(f) is a nonempty (distinguished) open subset…
user198206
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1 answer

Show that the Zariski topology on $A^2$ is not the product topology on $A^1 \times A^1$. (Hint: consider the diagonal.)

This is an Exercise in An Invitation to Algebraic Geometry by Karen Smith. I'm not sure what the hint means thus have no clue how to approach. Any thought please? Thanks very much!
nekodesu
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Coordinate ring of general linear group

Let $n$ be a positive integer and let $k$ be an algebraically closed field. What is the coordinate ring of $GL(n,k)$ (the set of all $n \times n$ matrices with entries in $k$)? Here we identify this set as a subset of $k^{n^{2}}$. Would it suffice…
user10
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Are there formal criteria for morphisms of stacks?

Let $f:X\to Y$ be a representable morphism of algebraic stacks over an algebraically closed field $k$. I wonder is there is a "formal criterion" for checking the properties formally smooth, unramified, étale. If $X,Y$ were schemes, we would phrase…
Brenin
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