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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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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1answer
157 views

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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0answers
192 views

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 } ...
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2answers
884 views

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 ...
5
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1answer
228 views

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 (...
4
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0answers
104 views

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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1answer
151 views

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 $D_0$ ...
2
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0answers
76 views

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 ...
2
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0answers
109 views

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 ...
2
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1answer
513 views

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 ...
2
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1answer
481 views

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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0answers
471 views

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: $$\...
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1answer
830 views

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&...