TY - CHAP

T1 - Quadratic solutions of quadratic forms

AU - Kollár, János

N1 - Funding Information:
M. Lieblich, N. Lubbes, R. Parimala, B. Poonen, M. Skopenkov and B. Sturmfels for comments, discussions and references. I learned a lot of the early history, especially the role of [Ber1907], from F. Russo. Partial financial support was provided by the NSF under grant number DMS-1362960.

PY - 2018

Y1 - 2018

N2 - The aim of this note is to study solutions of a homogeneous quadratic equation q(x0,…, xn) = 0, defined over a field k, where the xi are themselves homogeneous polynomials of some degree d in r + 1 variables. Equivalently, we are looking at rational maps from projective r-space Pr to a quadric hypersurface Q, defined over a field k. The space of maps of P1 to a quadric Q is stably birational to Q if d is even and to the orthogonal Grassmannian of lines in Q if d is odd. Most of the paper is devoted to obtaining similar descriptions for the spaces parametrizing maps, given by degree 2 polynomials, from P2 to quadrics. The most interesting case is 4-dimensional quadrics when there are 5 irreducible components. The methods are mostly classical, involving the Veronese surface, its equations and projections. In the real case, these results provide some of the last steps of a project, started by Kummer and Darboux, to describe all surfaces that contain at least 2 circles through every point.

AB - The aim of this note is to study solutions of a homogeneous quadratic equation q(x0,…, xn) = 0, defined over a field k, where the xi are themselves homogeneous polynomials of some degree d in r + 1 variables. Equivalently, we are looking at rational maps from projective r-space Pr to a quadric hypersurface Q, defined over a field k. The space of maps of P1 to a quadric Q is stably birational to Q if d is even and to the orthogonal Grassmannian of lines in Q if d is odd. Most of the paper is devoted to obtaining similar descriptions for the spaces parametrizing maps, given by degree 2 polynomials, from P2 to quadrics. The most interesting case is 4-dimensional quadrics when there are 5 irreducible components. The methods are mostly classical, involving the Veronese surface, its equations and projections. In the real case, these results provide some of the last steps of a project, started by Kummer and Darboux, to describe all surfaces that contain at least 2 circles through every point.

UR - http://www.scopus.com/inward/record.url?scp=85052216819&partnerID=8YFLogxK

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U2 - 10.1090/conm/712/14348

DO - 10.1090/conm/712/14348

M3 - Chapter

AN - SCOPUS:85052216819

T3 - Contemporary Mathematics

SP - 211

EP - 249

BT - Contemporary Mathematics

PB - American Mathematical Society

ER -