scholarly journals The universal Severi variety of rational curves on K3 surfaces

2012 ◽  
Vol 45 (1) ◽  
pp. 159-174 ◽  
Author(s):  
Michael Kemeny
Keyword(s):  
K3 Surfaces ◽  
Rational Curves ◽  
Severi Variety

scholarly journals Density of rational curves on $$K3$$ surfaces

2012 ◽  
Vol 356 (1) ◽  
pp. 331-354 ◽  
Author(s):  
Xi Chen ◽  
James D. Lewis
Keyword(s):  
K3 Surfaces ◽  
Rational Curves

scholarly journals Constructing rational curves on K3 surfaces

2011 ◽  
Vol 157 (3) ◽  
pp. 535-550 ◽  
Author(s):  
Fedor Bogomolov ◽  
Brendan Hassett ◽  
Yuri Tschinkel
Keyword(s):  
K3 Surfaces ◽  
Rational Curves

scholarly journals K3 surfaces, rational curves, and rational points

2010 ◽  
Vol 130 (7) ◽  
pp. 1470-1479 ◽  
Author(s):  
Arthur Baragar ◽  
David McKinnon
Keyword(s):  
K3 Surfaces ◽  
Rational Points ◽  
Rational Curves

scholarly journals Rational curves on K3 surfaces

2011 ◽  
Vol 188 (3) ◽  
pp. 713-727 ◽  
Author(s):  
Jun Li ◽  
Christian Liedtke
Keyword(s):  
K3 Surfaces ◽  
Rational Curves

scholarly journals Algebraic boundaries of Hilbert’s SOS cones

2012 ◽  
Vol 148 (6) ◽  
pp. 1717-1735 ◽  
Author(s):  
Grigoriy Blekherman ◽  
Jonathan Hauenstein ◽  
John Christian Ottem ◽  
Kristian Ranestad ◽  
Bernd Sturmfels
Keyword(s):  
Algebraic Geometry ◽  
K3 Surfaces ◽  
Sums Of Squares ◽  
Numerical Algebraic Geometry ◽  
Hankel Matrices ◽  
Severi Variety ◽  
The Difference ◽  
Rank Constraints ◽  
Extreme Rays

AbstractWe study the geometry underlying the difference between non-negative polynomials and sums of squares (SOS). The hypersurfaces that discriminate these two cones for ternary sextics and quaternary quartics are shown to be Noether–Lefschetz loci of K3 surfaces. The projective duals of these hypersurfaces are defined by rank constraints on Hankel matrices. We compute their degrees using numerical algebraic geometry, thereby verifying results due to Maulik and Pandharipande. The non-SOS extreme rays of the two cones of non-negative forms are parametrized, respectively, by the Severi variety of plane rational sextics and by the variety of quartic symmetroids.

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