Invariants of real rational symplectic 4-manifolds and lower bounds in real enumerative geometry

By Jean-Yves Welschinger. 2003

100 citations recorded by the source corpus.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f75d795526c35da17cf44b8f355baa74094a317f

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Patchworking singular algebraic curves, non-Archimedean amoebas and enumerative geometry (2002, background): “nd thereby conﬁrm the formulae suggested by Mikhalkin in [17]. Here we focus on a related problem of computing the Welschinger number χ ∆(p 1 ,...,p r ), introduced in a general symplectic setting in [34]. In our situation it is the number of real nodal irreducible curves passing through the given r”
Labeled floor diagrams for plane curves (2009, background): “Welschinger invariants and odd floor diagrams The Gromov–Witten numberNd,0 has a “real” counterpart, the Welschinger invariant Wd [43, 44] that counts real rational curves of degree d through generic 3d − 1 points on the real projective plane, each with a certain sign.”
Floor diagrams relative to a conic, and GW-W invariants of Del Pezzo surfaces (2014, background): “2The mass mRX(C) is the quantity originally considered by Welschinger in [Wel03].”
Lower bounds for real solutions to sparse polynomial systems (2004, background): “If Nd is the Kontsevich number of such complex rational curves [11] and Wd is Welschinger’s invariant, then logNd and logWd are each asymptotic to 3d log d.”
A tropical calculation of the Welschinger invariants of real toric Del Pezzo surfaces (2004, background): “One of the important questions of real enumerative geometry is: for a real algebraic surface Σ, how many real rational curves in an ample linear system |D| pass through a conjugation-invariant configuration of −KΣD − 1 distinct generic points in Σ? Similarly to the complex case, where the answer is ”
Enumerative tropical algebraic geometry in R^2 (2003, methodology): “This distinction was used in [26] in order to get a real curve counting invariant with respect to the initial configuration P ⊂ (R∗)2.”
Quantum Structures for Lagrangian Submanifolds (2007, result): “, x5? It would of course be also interesting to understand the relation of the above to the works of Welschinger [68, 69, 70] and of Solomon [65].”
Counting curves via lattice paths in polygons (2002, background): “Consider the path γ : [ 0 , 8 ] → ∆ 3 depicted on the extreme left of Fig.”
Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry (2003, result): “With the exception of this non-triviality, all these results have been announced in [14].”
Spinor states of real rational curves in real algebraic convex 3-manifolds and enumerative invariants (2003, background): “Note that a similar invariant and similar lower bounds have already been obtained in [14], [15] using real rational curves in real symplectic 4-manifolds.”
Welschinger invariant and enumeration of real rational curves (2003, background): “However, several technical details are presented in another way because of our choice to work with equations instead of parametrizations.”
Logarithmic equivalence of Welschinger and Gromov-Witten invariants (2004, background): “Welschinger [10, 11] of a way to attribute weights of ±1 to real solutions so that the number of solutions counted with weights is independent of the configuration of points; more precisely, a real collection w may even contain complex conjugated pairs of points, and then the result depends only on ”
A Caporaso-Harris type formula for Welschinger invariants of real toric Del Pezzo surfaces (2006, background): “) Due to the Welschinger theorem [10, 11] (and the genericity of the complex structure on Σ), the number”
Tropical Geometry and its applications (2006, unknown): “Yet, as it was suggested in [34], one may prescribe a sign ±1 to every such curve so that the sum Wd of these signs becomes invariant in the special case of g = 0.”
Welschinger invariants of real Del Pezzo surfaces of degree ≥ 3 (2011, background): “They were introduced in [24], [25] and count, with appropriate signs, the real rational pseudo-holomorphic curves which pass through given real collections of points in a given real rational symplectic four-fold.”