2016/10/04 by Saugata Basu, Antonio Lerario, Basu, Saugata +6 · 1 citation
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Data Management and Algorithms #FOS: Mathematics #Geometry and complex manifolds #Mathematical Dynamics and Fractals #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.1610.01205
openalex publication_date 2016/10/04 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
We derive a formula expressing the average number En of real lines on a\nrandom hypersurface of degree 2n-3 in \ℝ textrmPn in terms of\nthe expected modulus of the determinant of a special random matrix. In the case\nn=3 we prove that the average number of real lines on a random cubic surface\nin \ℝ textrmP3 equals: E3=6
sqrt2-3. Our technique can also\nbe used to express the number Cn of complex lines on a generic hypersurface\nof degree 2n-3 in \ℂ textrmPn in terms of the determinant of a\nrandom Hermitian matrix. As a special case we obtain a new proof of the\nclassical statement C3=27.\n We determine, at the logarithmic scale, the asymptotic of the quantity En,\nby relating it to Cn (whose asymptotic has been recently computed D.\nZagier). Specifically we prove that:
limn
to
infty
frac
log En
log\nCn=
frac12.\n Finally we show that this approach can be used to compute the number\nRn=(2n-3)!! of real lines, counted with their intrinsic signs, on a generic\nreal hypersurface of degree 2n-3 in \ℝ textrmPn.\n