Related papers: Profiles, linear spaces, and unirationality of com…

The purpose of this note is to show that the subvarieties of small degree inside a general hypersurface of large degree come from intersecting with linear spaces or other varieties.

Given a 0-dimensional affine K-algebra R=K[x_1,...,x_n]/I, where I is an ideal in a polynomial ring K[x_1,...,x_n] over a field K, or, equivalently, given a 0-dimensional affine scheme, we construct effective algorithms for checking whether…

Two cycles on a projective variety over an algebraically closed field are shown to be rationally equivalent if and only if their difference equals a difference of complete intersections of a certain kind. Some of Bloch's conjectures for…

We prove that the derived category of a smooth complete intersection variety is equivalent to a full subcategory of the derived category of a smooth projective Fano variety. This enables us to define some new invariants of smooth projective…

Unprojection theory is a philosophy due to Miles Reid, which becomes a useful tool in algebraic geometry for the construction and the study of new interesting geometric objects such as algebraic surfaces and 3-folds. In the present work we…

We compute the automorphism scheme of a generic odd dimensional $(2,2)$-complete intersection in characteristic $2$. This is the only case for complete intersections having a non-trivial identity component in automorphism schemes apart from…

We show that the set of families of smooth well-formed Fano weighted complete intersections admits a natural partition with respect to the variance $\mathrm{var}(X) = \mathrm{coind}(X) - \mathrm{codim}(X)$. Moreover, we obtain the…

We prove that if a smooth variety with non-positive canonical class can be embedded into a weighted projective space of dimension $n$ as a well formed complete intersection and it is not an intersection with a linear cone therein, then the…

A very general hypersurface of dimension $n$ and degree $d$ in complex projective space is rational if $d \leq 2$, but is expected to be irrational for all $n, d \geq 3$. Hypersurfaces in weighted projective space with degree small relative…

We prove the birational rigidity of Fano complete intersections of index 1 with a singular point of high multiplicity, which can be close to the degree of the variety. In particular, the groups of birational and biregular automorphisms of…

The aim of this paper is to provide an explicit basis of the miniversal deformation of a monomial curve defined by a free semigroup -- these curves make up a notable family of complete intersection monomial curves. First, we dispense a…

A class of models intended to be as minimal and structureless as possible is introduced. Even in cases with simple rules, rich and complex behavior is found to emerge, and striking correspondences to some important core known features of…

In this paper we study the degrees of irrationality of hypersurfaces of large degree in a complex projective variety. We show that the maps computing the degrees of irrationality of these hypersurfaces factor through rational fibrations of…

An almost complete intersection ideal can be seen as a $d$-sequence ideal with the minimal number of generators being one more than its height. In this paper, we give exact formulas for the regularity of powers of graded almost complete…

By studying the theory of rational curves, we introduce a notion of rational simple connectedness for projective homogeneous spaces. As an application, we prove that over a function field of an algebraic surface, a projective homogeneous…

In this survey article we introduce the notion of frontals, which provides a class of generalised submanifolds with singularities but with well-defined tangent spaces. We present a review of basic theory and known studies on frontals in…

We study rationality questions for Fano schemes of linear spaces on smooth complete intersections of two quadrics, especially over non-closed fields. Our approach is to study hyperbolic reductions of the pencil of quadrics associated to…

We give an explicit construction of a large subset of F^n, where F is a finite field, that has small intersection with any affine variety of fixed dimension and bounded degree. Our construction generalizes a recent result of Dvir and Lovett…

This note gives the complete projective classification of rational, cuspidal plane curves of degree at least 6, and having only weighted homogeneous singularities. It also sheds new light on some previous characterizations of free and…

A portrait is a combinatorial model for a discrete dynamical system on a finite set. We study the geometry of portrait moduli spaces, whose points correspond to equivalence classes of point configurations on the affine line for which there…