What are the possible singular fibers of an elliptic fibration over a higher dimensional base? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T15:29:41Z http://mathoverflow.net/feeds/question/32286 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/32286/what-are-the-possible-singular-fibers-of-an-elliptic-fibration-over-a-higher-dime What are the possible singular fibers of an elliptic fibration over a higher dimensional base? JME 2010-07-17T14:18:47Z 2010-07-21T13:21:06Z <p>An elliptic fibration is a proper morphism $Y\rightarrow B$ between varieties such that the fiber over a general point of the base $B$ is a smooth curve of genus one. It is often required for the morphism to be flat, but I am open minded about it. In order to save your valuable time, I will first summarize my understanding of the current status and will then ask my question.</p> <ol> <li><p>If the elliptic fibration admits a section, it is birationally equivalent to a (singular) Weierstrass model.</p></li> <li><p>The singular fibers of an <strong>elliptic surface</strong> have been classified by <strong>Kodaira</strong> and it is closely related to ADE extended Dynkin diagrams. There are 9 types of singular fibers including two infinite series. Kodaira classification also determines the monodromy around each singular fiber. Kodaira classification is also reproduced by <strong>Néron</strong> and there is a famous algorithm by Tate (called <strong>Tate's algorithm</strong>) which helps determining the type of singular fibers for a Weierstrass model. </p></li> <li><p>In the case of an <strong>elliptic threefold</strong>, <strong>Rick Miranda</strong> considers flat resolutions of Weierstrass models. He has a classification of singular fibers that results from his analysis. He looks at '<strong>collisions of singularities</strong>' at the intersection of two divisors of the discriminant locus. He assumes several strong conditions like normal crossing of the discriminant locus and a well defined $j$-invariant. He often blow-up the base to get rid of "bad collisions''. He classifies the fibers that appears over the remained "good collisions''. There are 7 non-trivial types of collisions and 5 of them lead to non-Kodaira fibers including an infinite family.</p></li> <li><p>Miranda's analysis has been generalized by <strong>Michael Szydlo</strong> in his Phd thesis at Harvard under the supervision of Barry Mazur. He considers collisions for higher dimensional elliptic fibrations. However, this part of thesis has not been published (but he has a published paper on the part of his thesis that deals with elliptic fibrations over non-perfect residue fields). He assumes the same conditions as Miranda and shows that in higher codimension, there are a finite number of types of non-Kodaira fibers resulting from collisions of singular fibers. Actually his list his similar to the one of Miranda except for one case where they do their blow-ups in different order and (therefore) end up with different fibers. </p></li> </ol> <p>So here is my question:</p> <blockquote> <p>What else is known about the possible fibers of a higher dimensional elliptic fibrations? Is there a hope to get a classification which does not assume normal crossing? If not, why? Non-trivial examples of non-Kodaira and non-Miranda fibers are also welcome and examples of non-flat fibration as well.</p> </blockquote> <p>The reason why I won't ask for normal crossing is because geometrically it is not a natural thing to ask. After all, the discriminant of a Weierstrass equation is of the form $\Delta=4 f^3+27 g^2$. This is not really an invitation to ask for normal crossing... Moreover, in many cases relevant for applications to string theory, the normal crossing condition will not hold.</p>