Related papers: Curve classes on conic bundle threefolds and appli…

Let $X$ be either a general hypersurface of degree $n+1$ in $\mathbb P^n$ or a general $(2,n)$ complete intersection in $\mathbb P^{n+1}, n\geq 4$. We construct balanced rational curves on $X$ of all high enough degrees. If $n=3$ or $g=1$,…

We show that a standard conic bundle over a minimal rational surface is rational and its Jacobian splits as the direct sum of Jacobians of curves if and only if its derived category admits a semiorthogonal decomposition by exceptional…

The main purpose of this paper is to introduce a new approach to study families of nodal curves on projective threefolds. Precisely, given $X$ a smooth projective threefold, $\E$ a rank-two vector bundle on $X$, $L$ a very ample line bundle…

We study twists of the Burkhardt quartic threefold over non-algebraically closed base fields of characteristic different from 2,3,5. We show they all admit quartic models in projective four-space. We identify a Galois-cohomological…

Let $X$ be a geometrically irreducible smooth projective curve, of genus at least three, defined over the field of real numbers. Let $G$ be a connected reductive affine algebraic group, defined over $\mathbb R$, such that $G$ is nonabelian…

We define and study the motive of the moduli stack of vector bundles of fixed rank and degree over a smooth projective curve in Voevodsky's category of motives. We prove that this motive can be written as a homotopy colimit of motives of…

Let $K$ be a local field of residue characteristic $p$. Let $C$ be a curve over $K$ whose minimal proper regular model has totally degenerate semi-stable reduction. Under certain hypotheses, we compute the prime-to-$p$ rational torsion…

In this paper we develop a bridge between model theory, geometric topology, and geometric group theory. In particular, we investigate the Ivanov Metaconjecture from the point of view of model theory, and more broadly we seek to answer the…

We investigate the universal Severi variety of rational curves on K3 surfaces, which parametrises irreducible rational curves in a fixed class on varying K3 surfaces of fixed genus. We investigate the conjecuted irreducibility of this space…

Classically, an indecomposable class $R$ in the cone of effective curves on a K3 surface $X$ is representable by a smooth rational curve if and only if $R^2=-2$. We prove a higher-dimensional generalization conjectured by Hassett and…

We show that an element $w$ of a finite Weyl group $W$ is rationally smooth if and only if the hyperplane arrangement $I$ associated to the inversion set of $w$ is inductively free, and the product $(d_1+1) \cdots (d_l+1)$ of the…

We introduce new obstructions to rationality for geometrically rational threefolds arising from the geometry of curves and their cycle maps.

We study rational curves on general Fano hypersurfaces in projective space, mostly by degenerating the hypersurface along with its ambient projective space to reducible varieties. We prove results on existence of low-degree rational curves…

In this paper we develop the formalism of rational complex Bezier curves. This framework is a simple extension of the CAD paradigm, since it describes arc of curves in terms of control polygons and weights, which are extended to complex…

Branched covers of orbit cylinders are the basic examples of holomorphic curves studied in symplectic field theory. Since all curves with Fredholm index one can never be regular for any choice of cylindrical almost complex structure, we…

Let $X$ be the product of two projective spaces and consider the general CICY threefold $Y$ in $X$ with configuration matrix $A$. We prove the finiteness part of the analogue of the Clemens' conjecture for such a CICY in low bidegrees. More…

We adapt for algebraically closed fields $k$ of characteristic greater than $2$ two results of Voisin, on the decomposition of the diagonal of a smooth cubic hypersurface $X$ of dimension $3$ over $\mathbb C$, namely: the equivalence…

For $G$ a split semi-simple group scheme and $P$ a principal $G$-bundle on a relative curve $X\to S$, we study a natural obstruction for the triviality of $P$ on the complement of a relatively ample Cartier divisor $D \subset X$. We show,…

We study rational cuspidal curves in Hirzebruch surfaces. We provide two obstructions for the existence of rational cuspidal curves in Hirzebruch surfaces with prescribed types of singular points. The first result comes from Heegaard--Floer…

Let $X$ be any smooth simply connected projective surface. We consider some moduli space of pure sheaves of dimension one on $X$, i.e. $\mhu$ with $u=(0,L,\chi(u)=0)$ and $L$ an effective line bundle on $X$, together with a series of…