Une courbe $C$ projective et lisse de genre $g$, non hyperelliptique, admet un plongement canonique dans un espace projectif ${\mathbb{P}}^{g-1}$. Un résultat classique affirme que l’idéal gradué $I_C$ des équations de $C$ dans ${\mathbb{P}}^{g-1}$ est engendré par ses éléments de degré $2$, sauf si $C$ admet certains systèmes linéaires très particuliers. Mark Green en a proposé il y a vingt ans une vaste généralisation, qui décrit la résolution minimale de $I_C$ en fonction de l’existence de systèmes linéaires spéciaux sur $C$. Claire Voisin vient de...

Floor diagrams are a class of weighted oriented graphs introduced by E. Brugallé and the second author. Tropical geometry arguments lead to combinatorial descriptions of (ordinary and relative) Gromov–Witten invariants of projective spaces in terms of floor diagrams and their generalizations. In a number of cases, these descriptions can be used to obtain explicit (direct or recursive) formulas for the corresponding enumerative invariants. In particular, we use this approach to enumerate rational...

In recent years, mirror symmetry for open strings has exhibited some new connections between symplectic and enumerative geometry (A-model) and complex algebraic geometry (B-model) that in a sense lie between classical and homological mirror symmetry. I review the rôle played in this story by matrix factorizations and the Calabi-Yau/Landau-Ginzburg correspondence.

Questo manoscritto è un'introduzione al concetto di formule del grado e a qualche loro applicazione. In esso si dà una formalizzazione di quello che si intenderà con formula del grado, vengono enunciati due esempi: uno cosiddetto di primo livello ed uno più generale. Successivamente si descrivono le componenti di queste formule: i numeri e gli ideali di ostruzione. Dopo un breve accenno alla dimostrazione, il testo si conclude con una sezione in cui si analizzano esplicitamente varietà algebriche...

On majore la dimension de l’ensemble des hypersurfaces de ${\mathbb{P}}^N$ dont l’intersection avec une variété projective intègre fixée n’est pas intègre. Les majorations obtenues sont optimales. Comme application, on construit, quand c’est possible, des hypersurfaces dont les intersections avec toutes les variétés d’une famille de variétés projectives intègres sont intègres. Le degré des hypersurfaces construites est explicite.

For low order jets, it is known how to construct meromorphic frames on the space of the so-called vertical $k$-jets$J_{\mathsf{\text{vert}}}^k\left(𝒳\right)$ of the universal hypersurface$$𝒳\subset {\mathbb{P}}^{n+1}\times {\mathbb{P}}^{\frac{(n+1+d)!}{\left(\right(n+1)!d!)}-1}$$parametrizing all projective hypersurfaces $X\subset {\mathbb{P}}^{n+1}\left(ℂ\right)$ of degree $d$. In 2004, for $k=n$, Siu announced that there exist two constants $c_n\ge 1$ and $c_n^{\prime }\ge 1$ such that the twisted tangent bundle$$T_{J_{\mathsf{\text{vert}}}^n\left(𝒳\right)}\otimes {𝒪}_{{\mathbb{P}}^{n+1}}\left(c_n\right)\otimes {𝒪}_{{\mathbb{P}}^{\frac{(n+1+d)!}{\left(\right(n+1)!d!)}-1}}\left(c_n^{\prime }\right)$$is generated at every point by its global sections. In the present article, we establish this property outside a certain exceptional algebraic subset $\Sigma \subset J_{\mathsf{\text{vert}}}^n\left(𝒳\right)$ defined by the vanishing of certain Wronskians,...