Minimal number of singular fibers in a Lefschetz fibration

There exists a (relatively minimal) genus g Lefschetz fibration with only one singular fiber over a closed (Riemann) surface of genus h iff g greater than or equal to 3 and h greater than or equal to 2. The singular fiber can be chosen to be reducible or irreducible. Other results are that every Dehn twist on a closed surface of genus at least three is a product of two commutators and no Dehn twist on any closed surface is equal to a single commutator.


Legendrian realization in convex Lefschetz fibrations and convex stabilizations
Akbulut, Selman; Arıkan, Mehmet Fırat (Walter de Gruyter GmbH, 2015-05-01)
We show that, up to a Liouville homotopy and a deformation of compact convex Lefschetz fibrations on W, any Lagrangian submanifold with trivial first de Rham cohomology group, embedded on a (symplectic) page of the (induced) convex open book on partial derivative W, can be assumed to be Legendrian in partial derivative W with the induced contact structure. This can be thought as the extension of Giroux's Legendrian realization (which holds for contact open books) for the case of convex open books. We also s...
On the moduli spaces of fiber bundles of curves of genus >= 2
Onsiper, H (Springer Science and Business Media LLC, 2000-11-02)
We determine the moduli spaces parametrizing analytic fiber bundles of curves of genus g greater than or equal to 2 over curves of genus g(b) > (g + 1)/2.
Automorphisms of the Hatcher-Thurston complex
Irmak, Elmas; Korkmaz, Mustafa (Springer Science and Business Media LLC, 2007-12-01)
Let S be a compact, connected, orientable surface of positive genus. Let HT(S) be the Hatcher-Thurston complex of S. We prove that Ant HT(S) is isomorphic to the extended mapping class group of S modulo its center.
On entire rational maps of real surfaces
Ozan, Yıldıray (The Korean Mathematical Society, 2002-01-01)
In this paper, we define for a component X-0 of a nonsingular compact real algebraic surface X the complex genus of X-0, denoted by g(C)(X-0), and use this to prove the nonexistence of nonzero degree entire rational maps f : X-0 --> Y provided that g(C)(Y) > g(C)(X-0), analogously to the topological category. We construct connected real surfaces of arbitrary topological genus with zero complex genus.
Seven, Ahmet İrfan (American Mathematical Society (AMS), 2019-07-01)
In the structure theory of cluster algebras, principal coefficients are parametrized by a family of integer vectors, called c-vectors. Each c-vector with respect to an acyclic initial seed is a real root of the corresponding root system, and the c-vectors associated with any seed defines a symmetrizable quasi-Cartan companion for the corresponding exchange matrix. We establish basic combinatorial properties of these companions. In particular, we show that c-vectors define an admissible cut of edges in the a...
Citation Formats
M. Korkmaz, “Minimal number of singular fibers in a Lefschetz fibration,” PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, pp. 1545–1549, 2001, Accessed: 00, 2020. [Online]. Available: