Related papers: Hyperplane Sections of Hypersurfaces

The linear orbit of a degree d hypersurface in $\mathbb{P}^n$ is its orbit under the natural action of PGL(n+1), in the projective space of dimension $N =\binom{n+d}{d} - 1$ parameterizing such hypersurfaces. This action restricted to a…

We give an upper bound for the number of points of a hypersurface over a finite field that has no lines on, in terms of the dimension, the degree, and the number of the elements of the finite field.

Let $d$ be a positive square-free integer $\equiv 3 \pmod{4}$ such that there is no invariant of the ideal class group $\mathbb{Q}\lbrack \sqrt{-d}\rbrack$ which is divisible by $4$. We prove an asymptotic formula for the number of immersed…

Using the method of moving frames we analyze the algebra of differential invariants for surfaces in three-dimensional affine geometry. For elliptic, hyperbolic, and parabolic points, we show that if the algebra of differential invariants is…

Let $k$ be an algebraically closed field and let $b$ and $n$ be integers with $n\geq 3$ and $1\leq b \leq n-1.$ Consider the moduli space $X$ of hypersurfaces in $\mathbb{P}^n_k$ of fixed degree $l$ whose singular locus is at least…

Let $k$ be a perfect field and let $X\subset {\mathbb P}^N$ be a hypersurface of degree $d$ defined over $k$ and containing a linear subspace $L$ defined over an algebraic closure $\overline{k}$ with $\mathrm{codim}_{{\mathbb P}^N}L=r$. We…

We use the "closed point sieve" to prove a variant of a Bertini theorem over finite fields. Specifically, given a smooth quasi-projective subscheme X of P^n of dimension m over F_q, and a closed subscheme Z in P^n such that Z intersect X is…

In this note, we study various measures of irrationality for hypersurfaces in projective spaces which were recently proposed by Bastianelli, De Poi, Ein, Lazarsfeld and Ullery. In particular, we answer the question raised by Bastianelli…

We recursively compute the Gromov-Witten invariants of the Hilbert scheme of two points in the plane. By studying the space of stable maps and computing virtual contributions, we use these invariants to enumerate hyperelliptic plane curves…

We consider closed hypersurfaces smoothly immersed in hyperbolic manifolds up to homotopy and commensurability. We prove that if a closed hyperbolic manifold $M$ contains a sequence of asymptotically geodesic hypersurfaces, then $\pi_1(M)$…

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…

For prime degree hypersurfaces of dimension at least 3, Mori asked if every smooth proper limit is still a hypersurface. Interestingly in dimensions 1 and 2, this is not the case. For example, Griffin constructed explicit families of…

Classically, an indecomposable class $R$ in the cone of effective curves on a K3 surface $X$ is representable by a smooth rational curve if and only if $R^2=-2$. We prove a higher-dimensional generalization conjectured by Hassett and…

The paper is a contribution to the conjecture of Kobayashi that the complement of a generic curve in the projective plane is hyperbolic, provided the degree is at least five. Previously the authors treated the cases of two quadrics and a…

We establish an analytic Hasse principle for linear spaces of affine dimension m on a complete intersection over an algebraic field extension K of Q. The number of variables required to do this is no larger than what is known for the…

Traditional algebraic geometric invariants lose some of their potency in positive characteristic. For instance, smooth projective hypersurfaces may be covered by lines despite being of arbitrarily high degree. The purpose of this…

We describe degenerations of projective plane curves to curves containing a fixed line $l$ as a component, and show that $H^1({\overline V}_{n,d,m}, {\Cal O} (r))=0, r \in{\Bbb Z}$, where $V_{n,d,m}\subset {\Bbb P}^N (N = n(n+3)/2)$ is the…

In (equi-)affine differential geometry, the most important algebraic invariants are the affine (Blaschke) metric h, the affine shape operator S and the difference tensor K. A hypersurface is said to admit a pointwise symmetry if at every…

The Hilbert scheme of projective 3-folds of codimension 3 or more that are linear scrolls over the projective plane or over a smooth quadric surface or that are quadric or cubic fibrations over the projective line is studied. All known such…

We prove an effective bound for the degree of a smooth divisor of a hypersurface of P^n, n>4 (projective space over an algebraically closed field of characteristic zero). Our result follows from a strong (since the degree of the divisor is…