We study families of rational curves on irreducible holomorphic symplectic varieties. We give a necessary and sufficient condition for a sufficiently ample linear system on a holomorphic symplectic variety of $K3^{[n]}$-type to contain a uniruled divisor covered by rational curves of primitive class. In particular, for any fixed $n$, we show that there are only finitely many polarization types of holomorphic symplectic variety of $K3^{[n]}$-type that do not contain such a uniruled divisor. As an application, we provide a generalization of a result due to Beauville–Voisin on the Chow group of $0$-cycles on such varieties.

Let $X$ be a smooth projective Fano variety over the complex numbers. We study the moduli space of rational curves on $X$ using the perspective of Manin’s conjecture. In particular, we bound the dimension and number of components of spaces of rational curves on $X$. We propose a geometric Manin’s conjecture predicting the growth rate of a counting function associated to the irreducible components of these moduli spaces.

We construct new indecomposable elements in the higher Chow group $CH^2(A,1)$ of a principally polarized Abelian surface over a $p$-adic local field, which generalize an element constructed by Collino [Griffiths’ infinitesimal invariant and higher K-theory on hyperelliptic Jacobians, J. Algebraic Geom. 6 (1997), 393–415]. These elements are constructed using a generalization, due to Birkenhake and Wilhelm [Humbert surfaces and the Kummer plane, Trans. Amer. Math. Soc. 355 (2003), 1819–1841 (electronic)], of a classical construction of Humbert. They can be used to prove a non-Archimedean analogue of the Hodge-${\mathcal{D}}$-conjecture – namely, the surjectivity of the boundary map in the localization sequence – in the case where the Abelian surface has good and ordinary reduction.

We discuss a possible approach to the study of the vanishing of the Kobayashi pseudo-metric of a projective variety X, using chains of rational or elliptic curves contained in an arbitrarily small neighbourhood of X in projective space for the Euclidean topology.

We show that minimal rational components on a complete toric manifold X correspond bijectively to some special primitive collections in the fan defining X, and the associated varieties of minimal rational tangents are linear subspaces. Two applications are given: the first is a classification of n-dimensional toric Fano manifolds with a minimal rational component of degree n, and the second shows that any complete toric manifold satisfying certain combinatorial conditions on the fan has the target rigidity property.

We 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 $n$.

We introduce the notion of families of $n$-marked, smooth, rational tropical curves over smooth tropical varieties and establish a one-to-one correspondence between (equivalence classes of) these families and morphisms from smooth tropical varieties into the moduli space of $n$-marked, abstract, rational, tropical curves ${{\mathcal{M}}_{n}}$.

Let $X$ be a smooth complex projective variety, and let $H\,\in \,\text{Pic}\left( X \right)$ be an ample line bundle. Assume that $X$ is covered by rational curves with degree one with respect to $H$ and with anticanonical degree greater than or equal to $\left( \dim\,X\,-\,1 \right)/2$. We prove that there is a covering family of such curves whose numerical class spans an extremal ray in the cone of curves $\text{NE}\left( X \right)$.

Let (X, cX) be a convex projective surface equipped with a real structure. The space of stable maps carries different real structures induced by cX and any order two element τ of permutation group Sk acting on marked points. Each corresponding real part ℝτ is a real normal projective variety. As the singular locus is of codimension bigger than two, these spaces thus carry a first Stiefel–Whitney class for which we determine a representative in the case k = c1(X)d − 1 where c1(X) is the first Chern class of X. Namely, we give a homological description of these classes in term of the real part of boundary divisors of the space of stable maps.

In this article we will show that there are infinitely many symmetric, integral 3 × 3 matrices, with zeros on the diagonal, whose eigenvalues are all integral. We will do this by proving that the rational points on a certain non-Kummer, singular $\text{K3}$ surface are dense. We will also compute the entire Néron–Severi group of this surface and find all low degree curves on it.

Using the $\sharp$-minimal model program of uniruled varieties we show that, for any pair $(X, {\mathcal H})$ consisting of a reduced and irreducible variety $X$ of dimension $k \geq 3$ and a globally generated big line bundle ${\mathcal H}$ on $X$ with $d:= {\mathcal H}^k$ and $n:= h^0(X, {\mathcal H})-1$ such that $d<2(n-k)-4$, then $X$ is uniruled of ${\mathcal H}$-degree one, except if $(k,d,n)=(3,27,19)$ and a ${\sharp}$-minimal model of $(X, {\mathcal H})$ is $({\mathbb P}^3,{\mathcal O}_{{\mathbb P}^3}(3))$. We also show that the bound is optimal for threefolds.

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

In the Mathematical Review of ‘On a conjecture of Kontsevich and variants of Castelnuovo's lemma’ [Compositio Mathematica 115 (1999), 205–230], Gizatullin pointed out that the proof as written is incomplete for the complex case because the possibility of a certain equation vanishing identically was ignored. This corrigendum addresses the missing case. Note that it is simply another iteration of the argument already present in the article.

We show that the vanishing order of a non-zero vector field at a generic point of a smooth Fano variety of Picard number 1 cannot exceed the dimension of the Fano variety. Furthermore, if there exist only finitely many rational curves of minimal degree through a generic point of the Fano variety, we show that a non-zero vector field cannot vanish at a generic point of the Fano variety.

Let A=(aij) be an orthogonal matrix (over R or C) with no entries zero. Let B= (bij) be the matrix defined by bij= 1/ai j. M. Kontsevich conjectured that the rank of B is never equal to three. We interpret this conjecture geometrically and prove it. The geometric statement can be understood as variants of the Castelnuovo lemma and Brianchon‘s theorem.