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{\Large המחלקה למתמטיקה, בן-גוריון}

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{\Huge גאומטריה אלגברית ותורת המספרים}\\[0.2\baselineskip]

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\textbf{ב}\emph{יום רביעי, 13 בדצמבר, 2017}
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\textbf{בשעה} \emph{15:10 -- 16:30}
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\textbf{ב}\emph{Math -101}

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ההרצאה

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{\Large\bfseries Discriminant of the ordinary transversal singularity type\par}
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תינתן על-ידי
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{\large\scshape Dmitry Kerner 
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  (BGU)
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\textbf{תקציר:}
  Singular spaces appear everywhere. And the singularity is often non-isolated, i.e. the singular locus is of positive dimension. These non-isolated singularities are more complicated and less studied.

Let X be a variety with singular locus Z, the simplest example being the surface \{y\^{}2=x\^{}2z\}. Generically along Z the singularity ''factorizes``, i.e. X is locally at each point the product:
(the germ of Z)\textbackslash{}times (the germ of space with an isolated singularity).

But at some special points of Z the picture degenerates and  the family of sections of X, transversal to Z, becomes not equi-singular (in whichever sense). These points form the discriminant   of transversal singularity type.  We study this discriminant, assuming X,Z are locally complete intersections and X is of ''ordinary type``  generically along Z.

First I will define the discriminant, as a subscheme of Z, and formulate its properties. This discriminant is a (effective) Cartier divisor in Z, nef but not necessarily ample, with nice pullback/pushforward properties under some maps. The discriminant deforms flatly under some deformations of X.

Then I will give the formula for the class of this discriminant in the cohomology/Chow group/Picard group of Z. This class ''counts the number of points`` where the transversal type degenerates as one travels along the singular locus.  In most cases this class is not zero (when Z is complete or projective). This places a ''topological`` obstruction to the naive expectation (from differential geometry): ''In the generic case the transversal type does not degenerate``.

The talk is based on  arXiv:1705.11013 and  arXiv:1308.6045.
  


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