There exists a function f : N → N such that for every positive integer d, every quasi-finite field K and every projective hypersurface X of degree d and dimension ≥ f(d), the set X(K) is non-empty. This is a special case of a more general result about intersections of hypersurfaces of fixed degree in projective spaces of sufficiently high dimension over fields with finitely generated Galois gro...

Let X be a smooth projective variety, let L be a very ample invertible sheaf on X and assume N+1 = dim(H(X,L)), the dimension of the space of global sections of L. Let P1, . . . , Pt be general points on X and consider the blowing-up π : Y → X of X at those points. Let Ei = π−1(Pi) be the exceptional divisors of this blowing-up. Consider the invertible sheaf M := π∗(L) ⊗ OY (−E1 − . . . − Et) o...

The number of apparent double points of a smooth, irreducible projective variety X of dimension n in P is the number of secant lines to X passing through the general point of P. This classical notion dates back to Severi. In the present paper we classify smooth varieties of dimension at most three having one apparent double point. The techniques developed for this purpose allow to treat a wider...

Contents 1 Conventions and basic notation 2 2 Hilbert variety: the general framework 3 3 The Hilbert curve 8 4 Cubic Hilbert curves 12 5 Image of the Hilbert curve in P 3 17 6 Fibrations and singular points of the Hilbert curve at infinity 19 7 Serre-invariant curves 23 Abstract Let X be a normal Gorenstein complex projective variety. We introduce the Hilbert variety V X associated to the Hilbe...

A classical theorem in complex algebraic geometry states that, for any smooth projective variety, the Gauss map is finite; in particular, a smooth variety and its Gauss image have the same dimension (with the obvious exception of a linear space). Furthermore, even when the variety is not smooth, Zak proved a lower bound on the dimension of its Gauss image in terms of the dimension of its singul...

Let \(({\mathcal {C}},{\mathbb {E}},{\mathfrak {s}})\) be an extriangulated category with a proper class \(\xi \) of \({\mathbb {E}}\)-triangles. In previous work, we introduced and studied the \)-\({\mathcal {G}}\)projective {G}}\)injective dimension for any object in \({\mathcal {C}}\). this paper, first give some characterizations by using derived functors on Consequently, show that followin...

In this paper we describe the facets cone associated to transversal polymatroid presented by A = {{1, 2}, {2, 3}, . . . , {n−1, n}, {n, 1}}. Using the Danilov-Stanley theorem to characterize the canonicale module, we deduce that the base ring associated to this polymatroid is Gorenstein ring. Also, starting from this polymatroid we describe the transversal polymatroids with Gorenstein base ring...

There exists a function f : N → N such that for every positive integer d, every quasi-finite field K and every projective hypersurface X of degree d and dimension ≥ f(d), the set X(K) is non-empty. This is a special case of a more general result about intersections of hypersurfaces of fixed degree in projective spaces of sufficiently high dimension over fields with finitely generated Galois gro...

Let R be a local ring of positive characteristic and X complex with nonzero finitely generated homology finite injective dimension. We prove that if the derived base change via Frobenius (or more generally, contracting) endomorphism has dimension then is Gorenstein. In particular, we give an affirmative answer to question by Falahola Marley [7, Question 3.9].