Editing Canonical bundle (section)

==The canonical bundle formula==
Let <math>X</math> be a normal surface. A '''genus <math>g</math> fibration''' <math>f:X\to B</math> of <math>X</math> is a [[Proper morphism|proper]] [[Flat morphism|flat]] morphism <math>f</math> to a smooth curve such that <math>f_*\mathcal{O}_X\cong \mathcal{O}_B</math> and all fibers of <math>f</math> have [[arithmetic genus]] <math>g</math>. If <math>X</math> is a smooth projective surface and the [[Fiber (mathematics)|fibers]] of <math>f</math> do not contain rational curves of self-intersection <math>-1</math>, then the fibration is called '''minimal'''. For example, if <math>X</math> admits a (minimal) genus 0 fibration, then is <math>X</math> is birationally ruled, that is, birational to <math>\mathbb{P}^1\times B</math>.

For a minimal genus 1 fibration (also called [[Elliptic surface|elliptic fibrations]]) <math>f:X\to B</math> all but finitely many fibers of <math>f</math> are geometrically integral and all fibers are geometrically connected (by [[Zariski's connectedness theorem]]). In particular, for a fiber <math>F=\sum^{n}_{i=1}a_iE_i</math> of <math>f</math>, we have that <math>F.E_i=K_X.E_i=0,</math> where <math>K_X</math> is a canonical divisor of <math>X</math>; so for <math>m=\operatorname{gcd}(a_i)</math>, if <math>F</math> is geometrically integral if <math>m=1</math> and <math>m>1</math> otherwise.

Consider a minimal genus 1 fibration <math>f:X\to B</math>. Let <math>F_1,\dots,F_r</math> be the finitely many fibers that are not geometrically integral and write <math>F_i=m_iF_i^'</math> where <math>m_i>1</math> is greatest common divisor of coefficients of the expansion of <math>F_i</math> into integral components; these are called '''multiple fibers'''. By [[Base change theorems|cohomology and base change]] one has that <math>R^1f_*\mathcal{O}_X=\mathcal{L}\oplus\mathcal{T}</math> where <math>\mathcal{L}</math> is an invertible sheaf and <math>\mathcal{T}</math> is a torsion sheaf (<math>\mathcal{T}</math> is supported on <math>b\in B</math> such that <math>h^0(X_b,\mathcal{O}_{X_b})>1</math>). Then, one has that 
:<math>\omega_X\cong f^*(\mathcal{L}^{-1}\otimes \omega_{B})\otimes \mathcal{O}_X\left(\sum^r_{i=1}a_iF_i'\right)</math>
where <math>0\leq a_i<m_i</math> for each <math>i</math> and <math>\operatorname{deg}\left(\mathcal{L}^{-1}\right)=\chi(\mathcal{O}_X)+\operatorname{length}(\mathcal{T})</math>.<ref>{{cite book |last=Badescu |first=Lucian |author-link=Lucian Badescu |date=2001 |title=Algebraic Surfaces |publisher=Springer Science & Business Media  |page=111 |isbn= 9780387986685}}</ref>
One notes that 
:<math>\operatorname{length}(\mathcal{T})=0\iff a_i=m_i-1</math>.

For example, for the minimal genus 1 fibration of a [[Hyperelliptic surface|(quasi)-bielliptic surface]] induced by the [[Albanese variety|Albanese morphism]], the canonical bundle formula gives that this fibration has no multiple fibers. A similar deduction can be made for any minimal genus 1 fibration of a [[K3 surface]]. On the other hand, a minimal genus one fibration of an [[Enriques surface]] will always admit multiple fibers and so, such a surface will not admit a section.