Surfaces of ℙ3 containing one conic and one line

We know from a result due to Noether–Lefschetz that a very general surface of degree at least 4 in [Formula: see text] contains only curves which are complete intersections with other surfaces. The main goal of this paper is to construct an explicit and smooth compactification of a parameter space for surfaces in [Formula: see text] of degree [Formula: see text] for all sufficiently large [Formula: see text], containing one conic and one line. The construction also applies to surfaces in [Formula: see text] containing one plane curve and one line. As an application, we compute the degree of the locus of surfaces of degree [Formula: see text] containing one conic and one line.

Principes locaux-globaux pour certaines fibrations en torseurs sous un tore

Letkbe a number field andTak-torus. Consider a family of torsors underT, i.e. a morphismf:X→ ℙ1kfrom a projective, smoothk-varietyXto ℙ1k, the generic fibreXη→ η of which is a smooth compactification of a principal homogeneous space underT⊗kη. We study the Brauer–Manin obstruction to the Hasse principle and to weak approximation forX, assuming Schinzel's hypothesis. We generalise Wei's recent results [21]. Our results are unconditional ifk=Qand all non-split fibres offare defined overQ. We also establish an unconditional analogue of our main result for zero-cycles of degree 1.

A compactification of the space of twisted cubics

We give an elementary, explicit smooth compactification of a parameter space for the family of twisted cubics. The construction also applies to the family of subschemes defined by determinantal nets of quadrics, e.g., cubic ruled surfaces in $\boldsymbol P^4$, Segre varieties in $\boldsymbol P^5$. It is suitable for applications of Bott's formula to a few enumerative problems.

On the comparison theorem for elementary irregular D-modules

Let U be a smooth quasi-projective variety over C and let f be a regular function on U. Let DU be the sheaf of algebraic differential operators on U and let M be a regular holonomic DU-module: here, regular means that there exists some smooth compactification X of U and some extension of M as a DX-module which is regular holonomic on X (one also may avoid the use of a smooth compactification to define regularity, see [17]).