330 research outputs found
Generalized isothermic lattices
We study multidimensional quadrilateral lattices satisfying simultaneously
two integrable constraints: a quadratic constraint and the projective Moutard
constraint. When the lattice is two dimensional and the quadric under
consideration is the Moebius sphere one obtains, after the stereographic
projection, the discrete isothermic surfaces defined by Bobenko and Pinkall by
an algebraic constraint imposed on the (complex) cross-ratio of the circular
lattice. We derive the analogous condition for our generalized isthermic
lattices using Steiner's projective structure of conics and we present basic
geometric constructions which encode integrability of the lattice. In
particular, we introduce the Darboux transformation of the generalized
isothermic lattice and we derive the corresponding Bianchi permutability
principle. Finally, we study two dimensional generalized isothermic lattices,
in particular geometry of their initial boundary value problem.Comment: 19 pages, 11 figures; v2. some typos corrected; v3. new references
added, higlighted similarities and differences with recent papers on the
subjec
Enumerative Real Algebraic Geometry
Enumerative Geometry is concerned with the number of solutions to a
structured system of polynomial equations, when the structure comes from
geometry. Enumerative real algebraic geometry studies real solutions to such
systems, particularly a priori information on their number. Recent results in
this area have, often as not, uncovered new and unexpected phenomena, and it is
far from clear what to expect in general. Nevertheless, some themes are
emerging.
This comprehensive article describe the current state of knowledge,
indicating these themes, and suggests lines of future research. In particular,
it compares the state of knowledge in Enumerative Real Algebraic Geometry with
what is known about real solutions to systems of sparse polynomials.Comment: Revised, corrected version. 40 pages, 18 color .eps figures. Expanded
web-based version at http://www.math.umass.edu/~sottile/pages/ERAG/index.htm
The KSBA compactification for the moduli space of degree two K3 pairs
Inspired by the ideas of the minimal model program, Shepherd-Barron,
Koll\'ar, and Alexeev have constructed a geometric compactification for the
moduli space of surfaces of log general type. In this paper, we discuss one of
the simplest examples that fits into this framework: the case of pairs (X,H)
consisting of a degree two K3 surface X and an ample divisor H. Specifically,
we construct and describe explicitly a geometric compactification
for the moduli of degree two K3 pairs. This compactification has a natural
forgetful map to the Baily-Borel compactification of the moduli space of
degree two K3 surfaces. Using this map and the modular meaning of ,
we obtain a better understanding of the geometry of the standard
compactifications of .Comment: 45 pages, 4 figures, 2 table
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