Ruled Fano fivefolds of index two

On the Griffiths Groups of Fano Manifolds of Calabi-Yau Hodge Type

A deep result of Voisin asserts that the Griffiths group of a general non-rigid Calabi-Yau (CY) 3-fold is infinitely generated. This theorem builds on an earlier method of hers which was implemented by Albano and Collino to prove the same result for a general cubic sevenfold. In fact, Voisin's method can be utilized precisely because the variation of Hodge structure on a cubic 7-fold behaves just like the variation of Hodge structure of a Calabi-Yau 3-fold. We explain this relationship concretely using Kontsevitch's noncommutative geometry. Namely, we show that for a cubic 7-fold, there is a n…

ruled Fano 3-折叠的 Picard 等级 2 和指数 1 的瞬子 sheaves

Instanton Sheaves on Ruled Fano 3-folds of Picard Rank 2 and Index 1

本文研究了 Picard 等级 2 和指数 1 的 ruled Fano 3-折叠上 rank 2 h-instanton sheaves 的存在性，并证明了 Ulrich bundles 的存在性。

Rational surfaces in index-one Fano hypersurfaces

We give the first evidence for a conjecture that a general, index-one, Fano hypersurface is not unirational: (i) a general point of the hypersurface is contained in no rational surface ruled, roughly, by low-degree rational curves, and (ii) a general point is contained in no image of a Del Pezzo surface.

Fano hypersurfaces with arbitrarily large degrees of irrationality

We show that complex Fano hypersurfaces can have arbitrarily large degrees of irrationality. More precisely, if we fix a Fano index e, then the degree of irrationality of a very general complex Fano hypersurface of index e and dimension n is bounded from below by a constant times $\sqrt{n}$. To our knowledge this gives the first examples of rationally connected varieties with degrees of irrationality greater than 3. The proof follows a degeneration to characteristic p argument which Kollár used to prove nonrationality of Fano hypersurfaces. Along the way we show that in a family of varieties, …

Rank 2 stable sheaves with odd determinant on Fano threefolds of genus 9

By the description due to Mukai and Iliev, a smooth prime Fano threefold X of genus 9 is associated to a surface P(V), ruled over a smooth plane quartic Gamma. We use Kuznetsov's integral functor to study rank-2 stable sheaves on X with odd determinant. For each c_2 \geq 7, we prove that a component of their moduli space M_X(2,1,c_2) is birational to a Brill-Noether locus of bundles on Gamma having enough sections when twisted by V. Moreover we prove that M_X(2,1,7) is isomorphic to the blowing-up of the Picard variety Pic^2(Gamma) along the curve parametrizing lines contained in X.

Classification of Mukai pairs with dimension $4$ and rank $2$

We give the complete classification of Mukai pairs of dimension $4$ and rank $2$ with Picard number one, that is, pairs $(X,E)$ where $X$ is a Fano $4$-fold with Picard number one, and $E$ is an ample vector bundle of rank two on $X$ with $c_1(X) = c_1(E)$. Equivalently, the present paper completes the classification of ruled Fano $5$-folds with index two, which was partially done by C. Novelli and G. Occhetta in 2007.

Strange duality on rational surfaces II: higher rank cases

We study Le Potier's strange duality conjecture on a rational surface. We focus on the strange duality map $SD_{c_n^r,L}$ which involves the moduli space of rank $r$ sheaves with trivial first Chern class and second Chern class $n$, and the moduli space of 1-dimensional sheaves with determinant $L$ and Euler characteristic 0. We show there is an exact sequence relating the map $SD_{c_r^r,L}$ to $SD_{c^{r-1}_{r},L}$ and $SD_{c_r^r,L\otimes K_X}$ for all $r\geq1$ under some conditions on $X$ and $L$ which applies to a large number of cases on $\p^2$ or Hirzebruch surfaces . Also on $\mathbb{P}^2…