2019/10/18 by Hannah Markwig, Thomas Markwig, Markwig, Hannah +5
Computer Science · Mathematics · #14N10 #14T05 (Primary) 51M20 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1910.08585
openalex publication_date 2019/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The family of complex projective surfaces in projective three space of degree\nd having precisely \δ nodes as their only singularities has codimension\n\δ in the linear system of surfaces of degree d for sufficiently large\nd and is of degree\nN\δ,complex(d)=(4(d-1)3)^\δ/\δ!+O(d3\δ-3). In\nparticular, this number is polynomial in d.\n By means of tropical geometry, we explicitly describe\n(4d3)^\δ/\δ!+O(d3\δ-1) surfaces passing through a suitable\ngeneric configuration of n= binomd+33-\δ-1 points in projective three\nspace. These surfaces are close to tropical limits which we characterize\ncombinatorially, introducing the concept of floor plans for multinodal tropical\nsurfaces. The concept of floor plans is similar to the well-known floor\ndiagrams (a combinatorial tool for tropical curve counts): with it, we keep the\ncombinatorial essentials of a multinodal tropical surface which are sufficient\nto reconstruct the surface.\n In the real case, we estimate the range for possible numbers of real\nmulti-nodal surfaces satisfying point conditions. We show that, for a special\nconfiguration w of real points, the number N\δ,real(d,w) of real\nsurfaces of degree d having \δ real nodes and passing through w is\nbounded from below by (\(3)/(2)d3)^\δ/\δ! +O(d3\δ-1).\n We prove analogous statements for counts of multinodal surfaces in P1\×\nP2 and P1\× P1\× P1.\n