In this work, we investigate the behaviour of the covering gonality of a very general hypersurface in a product of projective spaces. Inspired by the work of Bastianelli, Ciliberto, Flamini and Suppino in [BCFS19] which addresses the case of a hypersurface in a projective space, we establish a similar result for very general smooth hypersurfaces of sufficiently large bi-degree. More precisely, we show that the covering gonality of such hypersurfaces can be computed by viewing them as a family of hypersurfaces over a projective space. Then, the curves computing the covering gonality lie entirely within the fibres of one of the families. This rules out transversal curves from computing the covering gonality. In addition to this, we investigate the behaviour of the joint covering gonality as in [LM23] and establish a lower bound for bi-degree large enough.
This paper concerns the existence of curves with low gonality on smooth hypersurfaces of sufficiently large degree. It has been recently proved that if $X\subset \mathbb{P}^{n+1}$ is a hypersurface of degree $d\geq n+2$, and if $C\subset X$ is an irreducible curve passing through a general point of $X$, then its gonality verifies $\mathrm{gon}(C)\geq d-n$, and equality is attained on some special hypersurfaces. We prove that if $X\subset \mathbb{P}^{n+1}$ is a very general hypersurface of degree $d\geq 2n+2$, the least gonality of an irreducible curve $C\subset X$ passing through a general poi…
We study the complexity of birational self-maps of a projective threefold $X$ by looking at the birational type of surfaces contracted. These surfaces are birational to the product of the projective line with a smooth projective curve. We prove that the genus of the curves occuring is unbounded if and only if $X$ is birational to a conic bundle or a fibration into cubic surfaces. Similarly, we prove that the gonality of the curves is unbounded if and only if $X$ is birational to a conic bundle.
This note is an appendix to 'Measures of irrationality for hypersurfaces of large degree' by L. Ein, R. Lazarsfeld and B. Ullery. We prove an existence result for families of curves having low gonality, and lying on fundamental loci of first order congruences of lines in $\mathbb{P}^{n+1}$. As an application, we follow the ideas of the main paper, and we present a slight refinement of a theorem included in it. In particular, we show that given a very general hypersurface $X\subset \mathbb{P}^{n+1}$ of degree $d \geq 3n-2 \geq 7$, and a dominant rational map $f \colon X \dashrightarrow \mathbb{…
The purpose of this paper is to show that for a complete intersection curve $C$ in projective space (other than a few stated exceptions), any morphism $f: C \to \mathbb{P}^r$ satisfying $\text{deg}\, f^*\mathcal{O}_{\mathbb{P}^r}(1) <\text{deg}\, C$ is obtained by projection from a linear space. In particular, we obtain bounds on the gonality of such curves and compute the gonality of general complete intersection curves. We also prove a special case of one of the well-known Cayley-Bacharach conjectures posed by Eisenbud, Green, and Harris.
对任意光滑不可约射影曲线$X$，角性序列$\{d_r \;| \; r \in \mathbb N \}$是$X$的正整数不变量的严格递增序列.在大多数已知情况下，$d_{r+1}$并不比$d_r$大多少。在我们的术语中，这意味着数字$d_r$满足斜率不等式。这是本文的目的是研究的情况下，这是不正确的。我们给出的例子极值曲线在$\PP^r$，曲线上的一般K3-曲面在$\PP^r$和完整的交点在$\PP^3$。
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 with balanced normal bundle, and reprove some results on irreducibility of spaces of rational curves of low degree.
The numbers of $\mathbb{F}_q$-points of nonsingular hypersurfaces of a fixed degree in an odd-dimensional projective space are investigated, and an upper bound for them is given. Also we give the complete list of nonsingular hypersurfaces each of which realizes the upper bound. This is a natural generalization of our previous study of surfaces in projective $3$-space.
We study Gauss-Manin systems of non tame Laurent polynomial functions. We focuse on Hori-Vafa models, which are the expected mirror partners of the small quantum cohomology of smooth hypersurfaces in weighted projective spaces.
We use the cut and paste relation $[Y^{[2]}] = [Y][\mathbb{P}^m] + \mathbb{L}^2 [F(Y)]$ in $K_0(\text{Var}_k)$ of Galkin--Shinder for cubic hypersurfaces arising from projective geometry to characterize cubic hypersurfaces of sufficiently high dimension under certain numerical or genericity conditions. Removing the conditions involving the middle Betti number from the numerical conditions used extends the possible $Y$ to cubic hypersurfaces, complete intersections of two quadric hypersurfaces, or complete intersections of two quartic hypersurfaces. The same method also gives a family of other …
We investigate the stronger form of the Bogomolov-Gieseker inequality on smooth hypersurfaces in the projective space of any degree and dimension. The main technical tool is the theory of tilt-stability conditions in the derived category.
We show that the Hirzebruch-Milnor class of a projective hypersurface, which gives the difference between the Hirzebruch class and the virtual one, can be calculated by using the Steenbrink spectra of local defining functions of the hypersurface if certain good conditions are satisfied, e.g. in the case of projective hyperplane arrangements, where we can give a more explicit formula. This is a natural continuation of our previous paper on the Hirzebruch-Milnor classes of complete intersections.
We compute Hochschild cohomology of projective hypersurfaces starting from the Gerstenhaber-Schack complex of the (restricted) structure sheaf. We are particularly interested in the second cohomology group and its relation with deformations. We show that a projective hypersurface is smooth if and only if the classical HKR decomposition holds for this group. In general, the first Hodge component describing scheme deformations has an interesting inner structure corresponding to the various ways in which first order deformations can be realized: deforming local multiplications, deforming restrict…
In this paper, we will show that the Hesselink stratification of a Hilbert scheme of hypersurfaces is independent of the choice of Plücker coordinate and there is a positive relation between the length of Hesselink's worst virtual 1-parameter subgroup and multiplicity of a projective hypersurface.
We show that over an algebraically closed field of characteristic not equal to 2, homological projective duality for smooth quadric hypersurfaces and for double covers of projective spaces branched over smooth quadric hypersurfaces is a combination of two operations: one interchanges a quadric hypersurface with its classical projective dual and the other interchanges a quadric hypersurface with the double cover branched along it.
We employ the formalism of vanishing cycles and perverse sheaves to introduce and study the vanishing cohomology of complex projective hypersurfaces. As a consequence, we give upper bounds for the Betti numbers of projective hypersurfaces, generalizing those obtained by different methods by Dimca in the isolated singularities case, and by Siersma-Tibăr in the case of hypersurfaces with a $1$-dimensional singular locus. We also prove a supplement to the Lefschetz hyperplane theorem for hypersurfaces, which takes the dimension of the singular locus into account, and we use it to give a new proof…
We study various measures of irrationality for hypersurfaces of large degree in projective space and other varieties. These include the least degree of a rational covering of projective space, and the minimal gonality of a covering family of curves. The theme is that positivity properties of canonical bundles lead to lower bounds on these invariants. In particular, we prove that if X is a very general smooth hypersurface of dimension n and degree d \ge 2n+1, then any dominant rational mapping from X to projective n-space must have degree at least d-1. We also propose a number of open problems,…
Let $\mathscr{X}\to W$ be a flat family of generically irreducible hypersurfaces of degree $d\geq 2$ in $\PP^n$ with singular locus of dimension $t$, with $W$ unirational of dimension $r$. We prove that if $n$ is large enough with respect to $d$, $r$ and $t$, then $\mathscr{X}$ is unirational. This extends results in \cite {Pre, HMP}.
We determine the second fundamental form of a variation of Hodge Structure of a smooth projective hypersurface using the classical identification of the Hodge structure and the action of the infinitesimal variation of Hodge structure with pieces of the Jacobian Ideal of the hypersurface. This simple algebraic description also implies that, in this case, the variation of Hodge structure satisfies a new set of second order partial differential equations.Comments are welcome!
Let $\mathbb{K}$ be an algebraically closed field of characteristic zero and $\mathbb{G}_a$ be the additive group of $\mathbb{K}$. We say that an irreducible algebraic variety $X$ of dimension $n$ over the field $\mathbb{K}$ admits an additive action if there is a regular action of the group $\mathbb{G}_a^n = \mathbb{G}_a \times \ldots \times \mathbb{G}_a$ ($n$ times) on $X$ with an open orbit. In this paper we find all projective toric hypersurfaces admitting additive action.
