Real Rational Curves Research Articles

Enumeration of non-nodal real plane rational curves

Welschinger invariants enumerate real nodal rational curves in the plane or in another real rational surface. We analyze the existence of similar enumerative invariants that count real rational plane curves having prescribed non-nodal singularities and passing through a generic conjugation-invariant configuration of appropriately many points in the plane. We show that an invariant like this is unique: it enumerates real rational three-cuspidal quartics that pass through generically chosen four pairs of complex conjugate points. As a consequence, we show that through any generic configuration of four pairs of complex conjugate points, one can always trace a pair of real rational three-cuspidal quartics.

Tropical vertex and real enumerative geometry

AbstractWe show that the commutator relations in the refined tropical vertex group can be expressed via the enumeration of suitable real rational curves in toric surfaces.

Refined count of oriented real rational curves

We introduce a quantum index for oriented real curves inside toric varieties. This quantum index is related to the computation of the area of the amoeba of the curve for some chosen 2 2 -form. We then make a refined signed count of oriented real rational curves solution to some enumerative problem. This generalizes the 2017 results of Mikhalkin to higher dimension. Finally, we use the tropical approach to relate these new refined invariants to previously known tropical refined invariants.

An imaginary refined count for some real rational curves

In 2015, Mikhalkin introduced a refined count for the real rational curves in a toric surface which pass through a set consisting of real points and pairs of complex conjugated points chosen generically on the toric boundary of the surface. He then proved that the result of this refined count depends only on the number of pairs of complex conjugated points on each toric divisor. Using the tropical geometry approach and the correspondence theorem, we address the computation of the refined count when the pairs of complex conjugated points are chosen purely imaginary and belonging to the same component of the toric boundary. Despite the non-genericity, we relate this refined count for purely imaginary values to the refined invariant of Mikhalkin for generic values. That allows us to extend the relation between these classical refined invariants and the tropical refined invariants from Block–Göttsche.

WDVV-type relations for Welschinger invariants: Applications

We first recall Solomon’s relations for Welschinger invariants counting real curves in real symplectic fourfolds and the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV)-style relations for Welschinger invariants counting real curves in real symplectic sixfolds with some symmetry. We then explicitly demonstrate that, in some important cases (projective spaces with standard conjugations, real blowups of the projective plane, and two- and threefold products of the one-dimensional projective space with two involutions each), these relations provide complete recursions determining all Welschinger invariants from basic input. We include extensive tables of Welschinger invariants in low degrees obtained from these recursions with Mathematica. These invariants provide lower bounds for counts of real rational curves, including with curve insertions in smooth algebraic threefolds.

WDVV-type relations for disk Gromov–Witten invariants in dimension 6

The first author’s previous work established Solomon’s WDVV-type relations for Welschinger’s invariant curve counts in real symplectic fourfolds by lifting geometric relations over possibly unorientable morphisms. We apply her framework to obtain WDVV-style relations for the disk invariants of real symplectic sixfolds with some symmetry, in particular confirming Alcolado’s prediction for \({\mathbb {P}}^3\) and extending it to other spaces. These relations reduce the computation of Welschinger’s invariants of many real symplectic sixfolds to invariants in small degrees and provide lower bounds for counts of real rational curves with positive-dimensional insertions in some cases. In the case of \({\mathbb {P}}^3\), our lower bounds fit perfectly with Kollár’s vanishing results.

Relative enumerative invariants of real nodal del Pezzo surfaces

The surfaces considered are real, rational and have a unique smooth real $$(-2)$$ -curve. Their canonical class K is strictly negative on any other irreducible curve in the surface and $$K^2>0$$ . For surfaces satisfying these assumptions, we suggest a certain signed count of real rational curves that belong to a given divisor class and are simply tangent to the $$(-2)$$ -curve at each intersection point. We prove that this count provides a number which depends neither on the point constraints nor on deformation of the surface preserving the real structure and the $$(-2)$$ -curve.

Enumeration of real curves in CP2n−1 and a Witten–Dijkgraaf–Verlinde–Verlinde relation for real Gromov–Witten invariants

We establish a homology relation for the Deligne-Mumford moduli spaces of real curves which lifts to a WDVV-type relation for real Gromov-Witten invariants of real symplectic manifolds; we also obtain a vanishing theorem for these invariants. For many real symplectic manifolds, these results reduce all genus 0 real invariants with conjugate pairs of constraints to genus 0 invariants with a single conjugate pair of constraints. In particular, we give a complete recursion for counts of real rational curves in odd-dimensional projective spaces with conjugate pairs of constraints and specify all cases when they are nonzero and thus provide non-trivial lower bounds in high-dimensional real algebraic geometry. We also show that the real invariants of the three-dimensional projective space with conjugate point constraints are congruent to their complex analogues modulo 4.

Classification of real rational knots of low degree in the 3-sphere

In this paper, we classify, up to rigid isotopy, real rational knots of degrees less than or equal to [Formula: see text] in a real quadric homeomorphic to the 3-sphere. We also study their connections with rigid isotopy classes of real rational knots of low degree in [Formula: see text] and classify real rational curves of degree 6 in the 3-sphere with exactly one ordinary double point.

Qualitative aspects of counting real rational curves on real K3 surfaces

We study qualitative aspects of the Welschinger-like $\mathbb Z$-valued count of real rational curves on primitively polarized real $K3$ surfaces. In particular, we prove that with respect to the degree of the polarization, at logarithmic scale, the rate of growth of the number of such real rational curves is, up to a constant factor, the rate of growth of the number of complex rational curves. We indicate a few instances when the lower bound for the number of real rational curves provided by our count is sharp. In addition, we exhibit various congruences between real and complex counts.

Welschinger invariants of real del Pezzo surfaces of degree ≥ 2

We compute the purely real Welschinger invariants, both original and modified, for all real del Pezzo surfaces of degree ≥ 2. We show that under some conditions, for such a surface X and a real nef and big divisor class D ∈ Pic (X), through any generic collection of - DKX - 1 real points lying on a connected component of the real part ℝX of X one can trace a real rational curve C ∈ |D|. This is derived from the positivity of appropriate Welschinger invariants. We furthermore show that these invariants are asymptotically equivalent, in the logarithmic scale, to the corresponding genus zero Gromov–Witten invariants. Our approach consists in a conversion of Shoval–Shustin recursive formulas counting complex curves on the plane blown up at seven points and of Vakil's extension of the Abramovich–Bertram formula for Gromov–Witten invariants into formulas computing real enumerative invariants.

Missing sets in rational parametrizations of surfaces of revolution

Parametric representations do not cover, in general, the whole geometric object that they parametrize. This can be a problem in practical applications. In this paper we analyze the question for surfaces of revolution generated by real rational profile curves, and we describe a simple small superset of the real zone of the surface not covered by the parametrization. This superset consists, in the worst case, of the union of a circle and the mirror curve of the profile curve.

Chord Diagrams and Generic Real Rational Planar Curves of Degree 4

Chord Diagrams and Generic Real Rational Planar Curves of Degree 4

On Higher Genus Welschinger Invariants of del Pezzo Surfaces

The Welschinger invariants of real rational algebraic surfaces count real rational curves which represent a given divisor class and pass through a generic conjugation-invariant configuration of points. No invariants counting real curves of positive genera are known in general. We indicate particular situations, when Welschinger-type invariants counting real curves of positive genera can be defined. We also prove the positivity and give asymptotic estimates for such Weischinger-type invariants for several del Pezzo surfaces of degree $\ge2$ and suitable real nef and big divisor classes. In particular, this yields the existence of real curves of given genus and of given divisor class passing through any appropriate configuration of real points on the given surface.

Counting Real Rational Curves onK3 Surfaces

We provide a real analog of the Yau–Zaslow formula counting rational curves on K3 surfaces. But man is a fickle and disreputable creature and perhaps, like a chessplayer , is interested in the process of attaining his goal rather than the goal itself. Fyodor Dostoyevsky, Notes from the Underground.

μ-Bases for complex rational curves

We present a fast algorithm for finding a μ-basis for any rational planar curve that has a complex rational parametrization. We begin by identifying two canonical syzygies that can be extracted directly from any complex rational parametrization without performing any additional calculations. For generic complex rational parametrizations, these two special syzygies form a μ-basis for the corresponding real rational curve. In those anomalous cases where these two canonical syzygies do not form a μ-basis, we show how to quickly calculate a μ-basis by performing Gaussian elimination on these two special syzygies. We also present an algorithm to determine if a real rational planar curve has a complex rational parametrization. Examples are provided to illustrate our methods.

Pencils on real curves

AbstractWe consider coverings of real algebraic curves to real rational algebraic curves. We show the existence of such coverings having prescribed topological degree on the real locus. From those existence results we prove some results on Brill‐Noether Theory for pencils on real curves. For coverings having topological degree \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\underline{0}$\end{document} we introduce the covering number k and we prove the existence of coverings of degree 4 with prescribed covering number.

Counting real curves with passage/tangency conditions

We study the following question: given a set P of 3d-2 points and an immersed curve G in the real plane R^2, all in general position, how many real rational plane curves of degree d pass through these points and are tangent to this curve. We count each such curve with a certain sign, and present an explicit formula for their algebraic number. This number is preserved under small regular homotopies of a pair (P, G), but jumps (in a well-controlled way) when in the process of homotopy we pass a certain singular discriminant. We discuss the relation of such enumerative problems with finite type invariants. Our approach is based on maps of configuration spaces and the intersection theory in the spirit of classical algebraic topology.

Sur la première classe de Stiefel-Whitney de l’espace des applications stables réelles vers l’espace projectif

Moduli space of genus zero stable maps to the projective three-space naturally carries a real structure such that the fixed locus is a moduli space for real rational spatial curves with real marked points. The latter is a normal projective real variety. The singular locus being in codimension at least two, a first Stiefel-Whitney class is well defined. In this paper, we determine a representative for the first Stiefel-Whitney class of such real space when the evaluation map is generically finite. This can be done by means of Poincaré duals of boundary divisors.

Detecting real singularities of a space curve from a real rational parametrization

In this paper we give an algorithm that detects real singularities, including singularities at infinity, and counts local branches and multiplicities of real rational curves in the affine n -space without knowing an implicitization. The main idea behind this is a generalization of the D -resultant (see [van den Essen, A., Yu, J.-T., 1997. The D -resultant, singularities and the degree of unfaithfulness. Proc. Amer. Math. Soc. 25 (3), 689–695]) to n rational functions. This allows us to find all real parameters corresponding to the real singularities between the solutions of a system of polynomials in one variable.