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
3
votes
0 answers

dimension of fibres under proper morphisms

How can we prove true that: If $f$ is proper, then the set $T_d=\{s\in S\mid \dim f^{-1}(s) \geq d\}$ is closed? Is this true if we do not require $f$ to be proper? (I think the semicontinuity in the dimension of fibres look like Hartshorne Thm…
user93417
3
votes
0 answers

intersection dimension theorem for any smooth variety

In shafarevich's book "Basic algebraic geometry", there is an intersection dimension theorem as follows: Let $V$, $W$ be any two irreducible closed subvarieties of $\mathbb{A}^n$ (the affine space over an algebraic closed field $k$), if $V\cap W\neq…
Lan
  • 141
3
votes
1 answer

Structure sheaves in different point are not isomorphic?

Suppose $X$ is a smooth projective variety over $\mathbb{C}$. How can one understand that in $D(Coh(X))$, the structure sheaves corresponding to different points of $X$ are all non-isomorphic? Here by structure sheaf I mean that points get the usual…
Karsten
  • 233
3
votes
0 answers

About the calculation of cohomology groups on projective space

In Chapter III, section $5$ of Hartshorne, we want to calculate the cohomology groups of sheaves $\mathcal{O}_X(n)$ on a projective space $X=\mathbb{P}_A^r$, where $A$ is a noetherian ring. We are going to prove the following…
Fei Hu
  • 103
3
votes
0 answers

Finite surjective morphisms between integral algebraic varieties and universal injectivity.

This is inspired by https://math.stackexchange.com/questions/705979/purely-inseparable-morphisms-and-factorizations-of-a-morphism-of-finite-type that I asked previously . Let $k$ be an algebraically closed field. Let $f: X \rightarrow Y$ be a…
user101036
  • 818
3
votes
1 answer

Normal bundle of a line in a blow-up

Let $x\in X=\mathbb{P}^3$. Consider the blow-up $\widetilde{X}\to X$ of $x$ in $X$. Let $l\ni x$ be a line in $X$ and $\widetilde{l}$ is its strict transform in $\widetilde{X}$. How to prove that the normal bundle $N_{\widetilde{l}/\widetilde{X}}$…
Alex Nelson
  • 297
3
votes
2 answers

Short exact sequence of sheaves and intersection of curves

Let $C_1$ and $C_2$ be two (smooth rational) curves, $D=C_1+C_2$, $C_1\cap C_2=p$ (a single point). Then how can I show that there is a short exact sequence of sheaves $ 0\rightarrow \mathcal{O}_{C_2}(-p)\rightarrow \mathcal{O}_{D} \rightarrow…
Tim Gore
3
votes
1 answer

Ensuring I have a closed point

Hartshorne, Algebraic Geometry, Exercise II.3.20, reads (in part): Let $X$ be an integral scheme of finite type over a field $k$. (a) [Prove:] For any closed point $P \in X$, $\dim X = \dim \mathcal{O}_P$, where for rings we always mean the Krull…
Daniel McLaury
  • 24,656
3
votes
1 answer

Projection $\pi : (x,y) \mapsto x$ of $V(y^2 - g(x))$ where g is cubic extends to a regular map of the projective closure

I'm supposed to prove that the map $\pi : (x,y) \mapsto x$ of $X = V(y^2 - g(x)) \subset \mathbb A^2$ where $g$ is cubic extends to a regular map of the projective closure $\overline \pi : \overline X \to \mathbb P^1$. The projective closure of X (I…
fhyve
  • 2,495
3
votes
0 answers

Two equivaltent Definitions for Inflexion Point

In the lecture we had the following two definitions for inflexion point: Let $P = (a,b)$, $D = V(f)$ and $C = (a+\lambda t, b+\mu t) = C(t)$, $t \in k$. $C$ is called inflexion line of $D$ in $P$ if $mult_P(C,D)\geq 3$. Then point $P$ is called…
Luca
  • 1,616
3
votes
1 answer

Two non isomorphic algebraic varieties

Consider the two subsets: $X_1=\mathbb P_1(\mathbb C)\setminus\{0,1,\infty, e \}$ and $X_2=\mathbb P_1(\mathbb C)\setminus\{0,1,\infty, \pi \}$. They are two varieties in the sense of the first chapter of Hartshorne (I'd like to avoid the scheme…
Dubious
  • 13,350
  • 12
  • 53
  • 142
3
votes
1 answer

Does every Jacobian over $\overline{\mathbf{Q}}$ have everywhere good reduction?

Let $J$ be the Jacobian of a smooth projective connected curve of genus $g>1$ over the field $\overline{\mathbf{Q}}$ of algebraic numbers. Does $J$ have everywhere good reduction? I know that there are abelian varieties of any dimension which do not…
Gooz
  • 374
3
votes
1 answer

$Z\subset \mathbb{P}^n$ irreducible iff its pre-image in $\mathbb{A}^{n+1}-\{0\}$ irreducible

I'm having trouble with this question (it's a homework question). If $p:\mathbb{A}^{n+1}-\{0\} \rightarrow \mathbb{P}^n$ is the canonical projection and $Z\subset \mathbb{P}^n$ is closed, then $Z$ is irreducible if and only if $p^{-1}(Z)$ is…
user68193
3
votes
1 answer

Ampleness and global generation of divisors on smooth projective varieties.

Let $X$ be a smooth projective complex surface and $H$ an ample divisor on $X$. My main question is whether for any divisor $D$ we can say that eventually $mH+D$ will be globally generated. Secondly, I wonder if the viceversa holds as well.
Heitor Fontana
  • 3,676
3
votes
0 answers

Branch locus of a projection from hypersurface

Given an algebraically closed number field $k$, we consider the projective space $\mathbb P_{n+1}$, the hyperplane $H=\{X_{n+1}=0\}\simeq\mathbb P_n$ and a hypersurface $X$ defined by a polynomial $f(X_0,\ldots,X_{n+1})$. We define on $X$ the…
Wesley Barnard
  • 31
1 2 3
…
99 100