Related papers: Rational curves of degree at most 9 on a general q…

We prove the "strong form" of the Clemens conjecture in degree 10. Namely, on a general quintic threefold F in P^4, there are only finitely many smooth rational curves of degree 10, and each curve is embedded in F with normal bundle…

We prove that the incidence scheme of rational curves of degree 11 on quintic threefolds is irreducible. This implies a strong form of the Clemens conjecture in degree 11. Namely, on a general quintic threefold $F$ in $\mathbb{P}^4$, there…

We introduce and study a likely condition that implies the following form of Clemens' conjecture in degrees $d$ between 10 and 24: given a general quintic threefold $F$ in complex $\IP^4$, the Hilbert scheme of rational, smooth and…

We prove the following statement, predicted by Clemens' conjecture: A generic quintic threefold contains only finitely many smooth rational curves of degree 12.

Clemens' conjecture states that the the number of rational curve in a generic quintic threefold is finite. If it is false we prove that certain periods of rational curves in such a quintic threefold must vanish. Our method is based on a…

Let $X_0$ be a generic quintic threefold in projective space $\mathbf P^4$ over the complex numbers. For a fixed natural number $d$, let $R_d(X_0)$ be the open sub-scheme of the Hilbert scheme, parameterizing irreducible rational curves of…

We give some necessary conditions for a smooth irreducible curve $C\subset \mathbb{P}^4$ to be isolated in a smooth quintic threefold, and also find a lower bound for $h^1(\mathcal{N}_{C/{\mathbb{P}^4}})$. Combining these with beautiful…

This is a series of two papers in which we solve the Clemens conjecture: there are only finitely many smooth rational curves of each degree in a generic quintic threefold. In this first paper, we deal with a family of smooth Calabi-Yau…

Let $X$ be the product of two projective spaces and consider the general CICY threefold $Y$ in $X$ with configuration matrix $A$. We prove the finiteness part of the analogue of the Clemens' conjecture for such a CICY in low bidegrees. More…

In characteristic $p>0$ and for $q$ a power of $p$, we compute the number of nonplanar rational curves of arbitrary degrees on a smooth Hermitian surface of degree $q+1$ under the assumption that the curves have a parametrization given by…

According to a conjecture of H. Clemens, the dimension of the space of rational curves on a general projective hypersurface should equal the number predicted by a na\"ive dimension count. In the case of a general hypersurface of degree 7 in…

For each $1\leq n\leq6$ we present formulas for the number of $n-$nodal curves in an $n-$dimensional linear system on a smooth, projective surface. This yields in particular the numbers of rational curves in the system of hyperplane…

Consider the smooth projective models C of curves y^2=f(x) with f(x) in Z[x] monic and separable of degree 2g+1. We prove that for g >= 3, a positive fraction of these have only one rational point, the point at infinity. We prove a lower…

Let k be a finite field with characteristic exceeding 3. We prove that the space of rational curves of fixed degree on any smooth cubic hypersurface over k with dimension at least 11 is irreducible and of the expected dimension.

From a topological viewpoint, a rational curve in the real projective plane is generically a smoothly immersed circle and a finite collection of isolated points. We give an isotopy classification of generic rational quintics in…

We consider all genus 2 curves over Q given by an equation y^2 = f(x) with f a squarefree polynomial of degree 5 or 6, with integral coefficients of absolute value at most 3. For each of these roughly 200000 isomorphism classes of curves,…

Let $X \subset \mathbb{P}^{n}$ be an unramified real curve with $X(\mathbb{R}) \neq \emptyset$. If $n \geq 3$ is odd, Huisman conjectures that $X$ is an $M$-curve and that every branch of $X(\mathbb{R})$ is a pseudo-line. If $n \geq 4$ is…

We develop a new general method for computing the decomposition type of the normal bundle to a projective rational curve. This method is then used to detect and explain an example of a Hilbert scheme that parametrizes all the rational…

We prove some lower bounds on certain twists of the canonical bundle of a codimension-2 subvariety of a generic hypersurface in projective space. In particular we prove that the generic sextic threefold contains no rational or elliptic…

We develop a technique that allows us to prove results about subvarieties of general type hypersurfaces. As an application, we use a result of Clemens and Ran to prove that a very general hypersurface of degree (3n+1)/2 \leq d \leq 2n-3…