Related papers: Serre's question on thin sets in projective space
In this paper we construct new derived invariants with integral coefficients using the theory of motifs, and give several applications. Specifically, we obtain the following results: For complex algebraic surfaces, we prove that certain…
We compute the maximum number of rational points at which a homogeneous polynomial can vanish on a weighted projective space over a finite field, provided that the first weight is equal to one. This solves a conjecture by Aubry, Castryck,…
We study a geometric version of the dimension growth conjecture. While it is closely related in spirit to themes arising in geometric Manin's conjecture, it applies in greater generality and provides more uniform bounds. For an irreducible…
For any affine hypersurface defined by a complete symmetric polynomial in $k\geq 3$ variables of degree $m$ over the finite field $\mathbb{F}_{q}$ of $q$ elements, a special case of our theorem says that this hypersurface has at least…
Weil's theorem gives the most standard bound on the number of points of a curve over a finite field. This bound was improved by Ihara and Oesterl\'e for larger genus. Recently, Hallouin and Perret gave a new point of view on these bounds,…
We look at sequences of positive integers that can be realized as degree sequences of iterates of rational dominant maps of smooth projective varieties over arbitrary fields. New constraints on the degree growth of endomorphisms of the…
For an abelian category and a distinguished object with a graded endomorphism ring a necessary and sufficient criterion is given so that the category is equivalent to the abelian quotient of the category of finitely presented graded modules…
The generating functions of the Severi degrees for sufficiently ample line bundles on algebraic surfaces are multiplicative in the topological invariants of the surface and the line bundle. Recently new proofs of this fact were given for…
We prove graded bounds on the individual Betti numbers of affine and projective complex varieties. In particular, we give for each $p,d,r$, explicit bounds on the $p$-th Betti numbers of affine and projective subvarieties of $\mathrm{C}^k$,…
A basic problem in the study of algebraic morphisms is to determine which sets can be realised as the image of an endomorphism of affine space. This paper extends the results previously obtained by the first author on the question of…
Given a finite set, $X$, of points in projective space for which the Hilbert function is known, a standard result says that there exists a subset of this finite set whose Hilbert function is ``as big as possible'' inside $X$. Given a finite…
We describe dualities and complexes of logarithmic forms and differentials for central affine and corresponding projective arrangements. We generalize the Borel-Serre formula from vector bundles to sheaves on projective d-space with locally…
We work out details of the extrinsic geometry for two Hilbert schemes of some contemporary interest: the Hilbert scheme of two points on the projective plane and the dense open set parametrizing non-planar clusters in the punctual Hilbert…
We endow the set of complements of a fixed subspace of a projective space with the structure of an affine space, and show that certain lines of such an affine space are affine reguli or cones over affine reguli. Moreover, we apply our…
The notion of Weyl modules, both local and global, goes back to Chari and Pressley in the case of affine Lie algebras, and has been extensively studied for various Lie algebras graded by root systems. We extend that definition to a certain…
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…
Consider a component of the Hilbert scheme whose general point corresponds to a degree d genus g smooth irreducible and nondegenerate curve in a projective variety X. We give lower bounds for the dimension of such a component when X is P^3,…
Affine normal directions provide intrinsic affine-invariant descent directions derived from the geometry of level sets. Their practical use, however, has long been hindered by the need to evaluate third-order derivatives and invert tangent…
We prove an upper bound for the number of rational points of bounded height on irreducible affine hypersurfaces. More precisely, given an irreducible polynomial $f \in \mathbb{Z}[X_1, \dots, X_n]$, we prove an upper bound on the number of…
The fundamental theorem of affine geometry is a classical and useful result. For finite-dimensional real vector spaces, the theorem roughly states that a bijective self-mapping which maps lines to lines is affine. In this note we prove…