TY - JOUR

T1 - TROPICAL FLOOR PLANS AND ENUMERATION OF COMPLEX AND REAL MULTI-NODAL SURFACES

AU - Markwig, Hannah

AU - Markwig, Thomas

AU - Shaw, Kris

AU - Shustin, Eugenii

N1 - Publisher Copyright:
c 2021 University Press, Inc.

PY - 2022

Y1 - 2022

N2 - The family of complex projective surfaces in P3 of degree d having precisely δ nodes as their only singularities has codimension δ in the linear system |OP3 (d)| for sufficiently large d and is of degree Nδ,P3C(d) = (4(d − 1)3)δ/δ! + O(d3δ−3). In particular, Nδ,P3C(d) is polynomial in d. By means of tropical geometry, we explicitly describe (4d3)δ/δ! + O(d3δ−1) surfaces passing through a suitable generic configuration of n = (d+33) − δ − 1 points in P3. These surfaces are close to tropical limits which we characterize combinatorially, introducing the concept of floor plans for multinodal tropical surfaces. The concept of floor plans is similar to the well-known floor diagrams (a combinatorial tool for tropical curve counts): with it, we keep the combinatorial essentials of a multinodal tropical surface S which are sufficient to reconstruct S. In the real case, we estimate the range for possible numbers of real multi-nodal surfaces satisfying point conditions. We show that, for a special configuration w of real points, the number Nδ,P3R(d, w) of real surfaces of degree d having δ real nodes and passing through w is bounded from below by (32 d3)δ /δ! + O(d3δ−1).

AB - The family of complex projective surfaces in P3 of degree d having precisely δ nodes as their only singularities has codimension δ in the linear system |OP3 (d)| for sufficiently large d and is of degree Nδ,P3C(d) = (4(d − 1)3)δ/δ! + O(d3δ−3). In particular, Nδ,P3C(d) is polynomial in d. By means of tropical geometry, we explicitly describe (4d3)δ/δ! + O(d3δ−1) surfaces passing through a suitable generic configuration of n = (d+33) − δ − 1 points in P3. These surfaces are close to tropical limits which we characterize combinatorially, introducing the concept of floor plans for multinodal tropical surfaces. The concept of floor plans is similar to the well-known floor diagrams (a combinatorial tool for tropical curve counts): with it, we keep the combinatorial essentials of a multinodal tropical surface S which are sufficient to reconstruct S. In the real case, we estimate the range for possible numbers of real multi-nodal surfaces satisfying point conditions. We show that, for a special configuration w of real points, the number Nδ,P3R(d, w) of real surfaces of degree d having δ real nodes and passing through w is bounded from below by (32 d3)δ /δ! + O(d3δ−1).

UR - http://www.scopus.com/inward/record.url?scp=85131448667&partnerID=8YFLogxK

U2 - 10.1090/jag/774

DO - 10.1090/jag/774

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AN - SCOPUS:85131448667

SN - 1056-3911

VL - 31

SP - 261

EP - 301

JO - Journal of Algebraic Geometry

JF - Journal of Algebraic Geometry

IS - 2

ER -