Low-degree rational curves on hypersurfaces in projective spaces and their fan degenerations

2021 ◽  
Vol 225 (2) ◽  
pp. 106492
Author(s):  
Ziv Ran
Keyword(s):  
Projective Spaces ◽  
Rational Curves ◽  
Low Degree

A Characterization of Products of Projective Spaces

2006 ◽  
Vol 49 (2) ◽  
pp. 270-280 ◽  
Author(s):  
Gianluca Occhetta
Keyword(s):  
Projective Spaces ◽  
Rational Curves

AbstractWe give a characterization of products of projective spaces using unsplit covering families of rational curves.

scholarly journals Spaces of rational curves on complete intersections

2013 ◽  
Vol 149 (6) ◽  
pp. 1041-1060 ◽  
Author(s):  
Roya Beheshti ◽  
N. Mohan Kumar
Keyword(s):  
Complete Intersection ◽  
Rational Curves ◽  
Complete Intersections ◽  
Constant Dimension ◽  
Low Degree

AbstractWe prove that the space of smooth rational curves of degree $e$ on a general complete intersection of multidegree $(d_1, \ldots , d_m)$ in $\mathbb {P}^n$ is irreducible of the expected dimension if $\sum _{i=1}^m d_i \lt (2n+m+1)/3$ and $n$ is sufficiently large. This generalizes a result of Harris, Roth and Starr [Rational curves on hypersurfaces of low degree, J. Reine Angew. Math. 571 (2004), 73–106], and is achieved by proving that the space of conics passing through any point of a general complete intersection has constant dimension if $\sum _{i=1}^m d_i$ is small compared to $n$.

scholarly journals Stable Rationality of del Pezzo Fibrations of Low Degree Over Projective Spaces

2018 ◽  
Vol 2020 (23) ◽  
pp. 9075-9119 ◽  
Author(s):  
Igor Krylov ◽  
Takuzo Okada
Keyword(s):  
Projective Space ◽  
Three Dimensional ◽  
Projective Spaces ◽  
Canonical Divisor ◽  
Stable Rationality ◽  
Higher Dimensional ◽  
Low Degree ◽  
Del Pezzo

Abstract The main aim of this article is to show that a very general three-dimensional del Pezzo fibration of degrees 1, 2, and 3 is not stably rational except for a del Pezzo fibration of degree 3 belonging to explicitly described two families. Higher-dimensional generalizations are also discussed and we prove that a very general del Pezzo fibration of degrees 1, 2, and 3 defined over the projective space is not stably rational provided that the anti-canonical divisor is not ample.

scholarly journals Singular Curves of Low Degree and Multifiltrations from Osculating Spaces

2020 ◽  
Vol 2020 (21) ◽  
pp. 8139-8182 ◽  
Author(s):  
Jarosław Buczyński ◽  
Nathan Ilten ◽  
Emanuele Ventura
Keyword(s):  
Linear System ◽  
Smooth Curve ◽  
Rational Curves ◽  
Arithmetic Genus ◽  
Smooth Curves ◽  
Low Degree ◽  
Complete Linear System ◽  
Osculating Spaces ◽  
Special Case ◽  
Schubert Cycles

Abstract In order to study projections of smooth curves, we introduce multifiltrations obtained by combining flags of osculating spaces. We classify all configurations of singularities occurring for a projection of a smooth curve embedded by a complete linear system away from a projective linear space of dimension at most two. In particular, we determine all configurations of singularities of non-degenerate degree $d$ rational curves in $\mathbb{P}^n$ when $d-n\leq 3$ and $d<2n$. Along the way, we describe the Schubert cycles giving rise to these projections. We also reprove a special case of the Castelnuovo bound using these multifiltrations: under the assumption $d<2n$, the arithmetic genus of any non-degenerate degree $d$ curve in $\mathbb{P}^n$ is at most $d-n$.

scholarly journals Moduli Spaces of Point Configurations and Plane Curve Counts

2019 ◽  
Author(s):  
Markus Reineke ◽  
Thorsten Weist
Keyword(s):  
Recurrence Relation ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Plane Curve ◽  
Projective Spaces ◽  
Rational Curves ◽  
Euler Characteristics ◽  
Point Configurations

Abstract We identify certain Gromov–Witten invariants counting rational curves with given incidence and tangency conditions with the Euler characteristics of moduli spaces of point configurations in projective spaces. On the Gromov–Witten side, S. Fomin and G. Mikhalkin established a recurrence relation via tropicalization, which is realized on the moduli space side using Donaldson–Thomas invariants of subspace quivers.

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