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 -