Today’s menu: Geometry and resolution of singular algebraic surfaces

Abstract

Abstract. The courses are Triviality, Tangency, Transversality, Symmetry, Simplicity, Singularity. These characteristic local plates serve as our invitation to algebraic surfaces and their resolution. Please take a seat.

Characterizing normal crossing hypersurfaces

“…Remark 5. One can construct surfaces in C 3 with the cusp as singular locus by blowing down, see [FH10]. However, then the Jacobian ideal will not be radical.…”

“…Let D be the divisor in C 3 defined by h = xz(x + z − y 2 ). This divisor is called Tülle and is studied in more detail in [FH10]. Tülle consists of three components, which are smooth, intersect pairwise transversally and whose triple intersection is a point, see fig.…”

Noncommutative resolutions of discriminants

“…One can for example obtain X by successively blowing up singular points. This process can be quite intricate, see for example the dessert in this menu [FH10]. The preimage of {0} = Sing(X) in X will be denoted by E = n i=1 E i and is a union of irreducible curves E i on X (this holds for example because one can construct X as an embedded resolution of X ⊆ k 3 , which implies that the preimage of Sing X must be a normal crossing divisor).…”

Complex singularities and contact topology

“…The reader interested in proofs of Theorem 2.40 may consult Cutkosky's book [35] or Kollár's book [94]. For introductions to various techniques of resolution of singularities of surfaces, one may consult Lipman [110], Faber and Hauser [52] and Popescu-Pampu [164].…”