Welschinger invariant and enumeration of real rational curves

By I. Itenberg, Viatcheslav Kharlamov, Eugenii Shustin. 2003

114 citations recorded by the source corpus.

Welschinger’s invariant bounds from below the number of real rational curves through a given generic collection of real points in the real projective plane. We estimate this invariant using Mikhalkin’s approach which deals with a corresponding count of tropical curves. In particular, our estimate implies that, for any positive integer d, there exists a real rational curve of degree d through any collection of 3d − 1 real points in the projective plane, and, moreover, asymptotically in the logarithmic scale at least one third of the complex plane rational curves through a generic point collection are real. We also obtain similar results for curves on other toric Del Pezzo surfaces.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Information Systems

View Paper: https://www.semanticscholar.org/paper/2f742745c4082ae4ae6e15c04866f3981402e8e0

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Citations in the corpus

Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry (2003, background): “Remark 2.4. Some lower bounds for the invariant χ d c1(X )d−1 have been obtained in [ 5 ] when X is a toric Del Pezzo surface.”
The Tropical Rank of a Tropical Matrix (2006, background): “that of Develin, Santos and Sturmfels [3]. The notion of tropical mathematics was introduced only in the past decade [5, 26], since then this theory has developed rapidly and led to many applications [4, 6, 13, 15, 16, 21, 22, 23, 24]; a short brief can be found in [14]. This mathematics takes place”
Logarithmic equivalence of Welschinger and Gromov-Witten invariants (2004, methodology): “A G ] 1 1 Ju l 2”
On the tropical Torelli map (2009, background): “For instance, this program has been carried out successfully for many problems of real and complex enumerative geometry, see for example [32,40,50,24,25,11,33,7,21].”
Lower bounds for real solutions to sparse polynomial systems (2004, background): “x) rational cubics passing through 8 real points in the plane, at least 8 are real [3, Prop. 4.7.3]. This was generalized by Welschinger [25], Mikhalkin [14, 15], and Itenberg, Kharlamov, and Shustin [9], to rational curves passing through real points on toric surfaces. Welschinger discovered an inv”
Enumeration of curves via floor diagrams (2007, unknown): “[3] Itenberg, I. and Kharlamov, V. and Shustin, E., Welschinger invariant and enumeration of real rational curves, Int.”
Welschinger invariants of real Del Pezzo surfaces of degree ≥ 3 (2011, methodology): “As we proved in [10, 11, 13, 14, 15], if X is either the plane blown up at a real points and b pairs of complex conjugate points, where a+2b ≤ 6, b ≤ 1, or a minimal two-component real conic bundle over P(1), or a two-component real cubic surface, then the (modified) Welschinger invariants of X are ”
A Caporaso-Harris type formula for Welschinger invariants of real toric Del Pezzo surfaces (2006, unknown): “Then, the −→s -peeling l−→s (∆) of ∆ is the convex hull of (∆ ∩ Z 2)\L−→s .”
Tropical Geometry and its applications (2006, unknown): “This theorem is the only currently known way to compute the Welschinger numbers, see [11].”
Spinor states of real rational curves in real algebraic convex 3-manifolds and enumerative invariants (2003, background): “Are the upper and lower bounds given by this corollary sharps ? How to compute the invariant χ p r ? See [13] for a discussion of related problems in real enumerative geometry and [5] for an estimation of the similar invariants constructed in [14], [15].”
On Block-Goettsche multiplicities for planar tropical curves (2012, background): “While his signs make perfect sense for real curves in any genus, the corresponding algebraic number of curves is not an invariant if g > 0 (see [9]).”
First steps in tropical geometry (2003, background): “d diﬀerent names for tropical varieties: logarithmic limit sets, Bergman fans, Bieri-Groves sets, and non-archimedean amoebas. All of these notions are essentially the same. Recent references include [5, 6, 7, 14, 17, 19]. For the relationship to Maslov dequantization see [20]. 2. Algebraic deﬁnitio”