arXiv subjects

Luca Tasin

Publications and source records attributed to Luca Tasin.

24 records · Page 2Linked to original sources

arXiv2015

Some low-dimensional hypersurfaces that are not stably rational

Using Voisin's method we prove that a very general hypersurface of degree at least 4 in complex projective space of dimension 6, 7, 8 or 9 is not stably rational and so, in particular, not rational. We obtain the same conclusion for the double covering of projective space of dimension 6, 7, 8 or 9, branched along a very general quartic hypersurface. On the other hand, such double coverings as well as general quartic hypersurfaces of dimension at least 5 are known to be unirational.

math.AG↗

arXiv2015

A remark on the Ueno-Campana's threefold

We show that the Ueno-Campana's threefold cannot be obtained as the blow-up of any smooth threefold along a smooth centre, answering negatively a question raised by Oguiso and Truong.

math.AG↗

arXiv2015

Fano varieties in Mori fibre spaces

We show that being a general fibre of a Mori fibre space is a rather restrictive condition for a Fano variety. More specifically, we obtain two criteria (one sufficient and one necessary) for a Q-factorial Fano variety with terminal singularities to be realised as a fibre of a Mori fibre space, which turn into a characterisation in the rigid case. We apply our criteria to figure out this property up to dimension three and on rational homogeneous spaces. The smooth toric case is studied and an interesting connection with K-semistability is also investigated.

math.AG↗

arXiv2015

Local Fano-Mori contractions of high nef-value

Let $X$ be a variety with at most terminal $\mathbb Q$-factorial singularities of dimension $n$. We study local contractions $f:X\to Z$ supported by a $\mathbb Q$-Cartier divisor of the type $K_X+ τL$, where $L$ is an $f$-ample Cartier divisor and $τ\geq 0$ is a rational number. Equivalently, $f$ is a Fano-Mori contraction associated to an extremal face in $\overline {NE(X)}_{K_X+τL = 0}$; these maps naturally arise in the context of the minimal model program. We prove that, if $τ> (n-3) >0$, the general element $X' \in |L|$ is a variety with at most terminal singularities. Then we apply this to characterize, via an inductive argument, some birational contractions as above with $τ> (n-3)\geq 0$.

math.AG↗

arXiv2013

Fano-Mori contractions of high length on projective varieties with terminal singularities

Let X be a projective variety with terminal singularities and let L be an ample Cartier divisor on X. We prove that if f is a birational contraction associated to an extremal ray $ R \subset \bar {NE(X)}$ such that R.(K_X+(n-2)L)<0, then f is a weighted blow-up of a smooth point. We then classify divisorial contractions associated to extremal rays R such that R.(K_X+rL)<0, where r is a non-negative rational number, and the fibres of f have dimension less or equal to r+1.

math.AG↗

arXiv2009

Koszul cohomology and singular curves

We investigate Koszul cohomology on irreducible nodal curves. In particular, we prove both Green and Green-Lazarsfeld conjectures for the general k-gonal nodal curve.

math.AG↗