Questions tagged [kodaira-dimension]
For questions about the Kodaira dimension of a compact complex manifold $X$, a numerical invariant which takes value in $\{-\infty, 0, 1, \dots, \dim X\}$.
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Kodaira dimension of symmetric products of curves
What is the Kodaira dimension of symmetric products of curves? That is, given a projective smooth, connected complex curve $C$, what is the Kodaira dimension of $C^{(d)}=C^d/\mathfrak S_d$?
When $d&...
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On a kind of Hilbert irreducibility theorem
Let us work over a number field $k$. Let $C$ be a non-empty open subscheme of $\mathbb{P}^{1}_{k}$, and $X\to C$ a family of smooth, projective hyperbolic curves such that $X(k)\to C(k)$ is surjective....
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Understanding what it means to be ''of general type''
I'm attempting to understand the Bombieri-Lang Conjecture:
If $X$ is a smooth projective variety of general type defined over a number field, then the set of rational points of $X$ is not dense.
I ...
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Properly elliptic surface with no multiple fibers and without a section
I am aware that if an elliptic surface contains multiple fibers, then it has no section. Is the converse false?
In particular, I am looking for an example of a projective, properly elliptic surface (...
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Calculate Kodaira dimension of a singular hypersurface
For a smooth projective hypersurface $H \subseteq \mathbb{P}^n$ of degree $d$ one can calculate its Kodaira dimension $\kappa(H)$, and find
$$\kappa(H) =
\begin{cases}
-\infty \qquad &\mbox{if } ...
3
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What are the possibilities of the general fibres in an Iitaka fibration?
This question is motivated by complex algebraic geometry.
If $X$ is a complex algebraic variety with Kodaira dimension in $[1,\dim X-1]$, then the Iitaka fibration (the rational map induced by the ...
3
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A question on the Kodaira dimension of 3-folds
Let $X$ a smooth projective $3$-fold. Assume that $X$ admits a finite rational map $f:X\dashrightarrow Y$ where $Y$ is a smooth Calabi-Yau 3-fold, and a fibration $g:X\rightarrow \mathbb{P}^2$ with a ...
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Rational curves on the image of the pluricanonical maps
Let $X$ be a compact complex manifold with canonical bundle $K_X$. Assume the Kodaira dimension $\kappa(X)$ is positive (but not maximal, i.e., $X$ is not of general type). Let $\varphi_m : X \...
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Kodaira dimension of the moduli space of curves
It is known that the moduli space $\overline{M}_{g}$ of genus $g$ curves is of general type for $g\geq 24$.
By Theorem 2.4 of
Logan, Adam The Kodaira dimension of moduli spaces of curves with ...
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On complex surfaces with Kodaira dimension 1
Let $S$ be a complex surface of Kodaira dimension $1$ and $\pi_{1}(S) \neq 1 $.
What is known on possible diffeomorphism types of such $complex$ surfaces with a given fundamental group? Is it true ...
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Kodaira dimension of spaces of rational curves in hypersurfaces
Let $X\subset\mathbb{P}^n$ be a general hypersurface of degree $d\leq n$, and $\overline{\mathcal{M}}_{0,0}(X,a)$ the Kontsevich space of degree $a$ rational curves in $X$.
Does there exist an ...
2
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Minimal model vs canonical model of a surface
When I have a projective surface $X$, for simplicity smooth, I can find a simpler smooth surface on its binational class. In this way we find in a finite number of steps the simplest surface $Y$, i.e. ...
2
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Log Kodaira dimension of Briekorn varities
Is there any formula or estimate of the log-Kodaira dimension of the Brieskorn variety $V_{a_0,\ldots,a_n}:=\{x_0^{a_0}+\ldots + x_n^{a_n}=1\}$ for $2\le a_0\le \ldots \le a_n$.
In particular, I ...
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Does anyone know if there is a generalization of symplectic Kodaira dimension beyond 4-manifolds?
I'm aware that in algebraic geometry, one has the Kodaira-Iitaka dimension, which generalizes the Kodaira dimension, but does anyone know if a correspondent generalization in the symplectic category ...
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first chern class versus compactifying divisor in Ramanujam's surface
I have an elementary question about Ramanujam's surface. Ramanujam's surface is naturally the complement of a singular divisor $D$ in the one point blow up of $CP^2$, $\mathbb{F}_1$. One can resolve ...
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Étale morphism over unirational/uniruled variety
Suppose we have an étale morphism between smooth quasi-projective (complex) varieties $X \rightarrow Y$ and assume that $Y$ is unirational. I am wondering whether we can somehow deduce that $X$ is ...
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Big divisors in family
Given a family of divisors $D_t$ on varieties $X_t$, there are examples that show that bigness is not well behaved (e.g. example 2.2.13 in Positivity 1, shows we can have a special fiber where $D_0$ ...
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Beauville Exercise VII.7 (3)-A proof that $\kappa(X)\geq \kappa(Y)$ for $f\colon X\to Y$ surjective morphism of smooth projective varieties
Here $\kappa(X)$ denotes the Kodaira dimension of a smooth projective variety $X$.
Question 1:
I would like to solve Exercise VII.7 (3) from the Beauville book "Complex Algebraic Surfaces":
...
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Subadditivity of Kodaira dimension
Given an algebraic fiber space $X \to B$ where $X$ and $B$ are smooth projective varieties over $\mathbb{C}$, it is known that the Kodaira dimensions satisfy the following subadditivity property:
$$\...