# 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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### 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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### 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. ...
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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 ...
1 vote
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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 ...
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 } ... 9 votes 2 answers 1k 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 votes 1 answer 316 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 votes 0 answers 110 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 ... 1 vote 1 answer 206 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 votes 0 answers 100 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 votes 0 answers 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 votes 1 answer 565 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 votes 1 answer 589 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 ... 1 vote 0 answers 527 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:$$\...
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&...