Minicourse in Real Enumerative Geometry
Monday April 17th, 2017
Time: 4:00 PM
Title: Real Algebraic Enumerative Geometry: Invariant Counts and Their Qualitative Properties.
Speaker: V.M.Kharlamov, University of Strasbourg
Location: Math Tower P-131
$\hspace{3mm}$Enumerative geometry is a classical part of algebraic geometry concerned with counting geometric figures subject to constraints imposed by other, fixed, figures. A fundamental property which makes the classical enumerative geometry to work is that the result does not depend on precise constraints data, provided the constraints are in a sufficiently general position and the ground field is algebraically closed.
$\hspace{3mm}$Over the field of real numbers the count of real solutions fails to be independent on the constraints. Therefore it was a common believe that enumerative geometry is limited to be essentially over algebraically closed fields. Thus, enumerative geometry over the reals remained practically non-existent up to the discovery by J.~Y.~Welschinger of an invariant integer valued signed count in a real version of one of the foundational enumerative problems of Gromov-Witten theory: counting real rational curves on real rational symplectic 4-manifolds constrained by an appropriate number of real points.
$\hspace{3mm}$Thanks to the invariance property such a count allowed not only to get non-trivial existence results, but, to a common surprise, to disclose a phenomenon of abundance: in the logarithmic scale the number of real solutions is asymptotically equivalent to the number of all complex solutions. The further developing in real symplectic, as well as algebraic, enumerative geometry has shown a strong persistence of such an abundance phenomenon and of some other qualitative properties of the underlying invariant counts.
$\hspace{3mm}$To present some typical results, and the methods employed, we will focus our discussion on two particular problems: counting real lines and higher dimensional linear subspaces on projective hypersurfaces and complete intersections (a typical problem from a kind of Real Schubert Calculus); and counting real rational curves on K3 surfaces (a typical problem from a kind of Real Gromov-Witten Calculus).
$\hspace{3mm}$In the first lecture I intend to present the general picture, including elementary definitions of the signs involved and the main statements, and to illustrate approaches developed so far.