Fano Hypersurfaces Research Articles

A categorical Torelli theorem for hypersurfaces

AbstractLet be a smooth Fano hypersurface of dimension and degree . The derived category of coherent sheaves on contains an interesting subcategory called the Kuznetsov component . We show that this subcategory, together with a certain autoequivalence called the rotation functor, determines uniquely if or if and . This generalizes a result by Huybrechts and Rennemo, who proved the same statement under the additional assumption that divides .

Fano hypersurfaces in positive characteristic

Fano hypersurfaces in positive characteristic

Rational endomorphisms of Fano hypersurfaces

Rational endomorphisms of Fano hypersurfaces

Metric SYZ conjecture for certain toric Fano hypersurfaces

Metric SYZ conjecture for certain toric Fano hypersurfaces

The Hasse principle for random Fano hypersurfaces

It is known that the Brauer--Manin obstruction to the Hasse principle is vacuous for smooth Fano hypersurfaces of dimension at least $3$ over any number field. Moreover, for such varieties it follows from a general conjecture of Colliot-Thélène that the Brauer--Manin obstruction to the Hasse principle should be the only one, so that the Hasse principle is expected to hold. Working over the field of rational numbers and ordering Fano hypersurfaces of fixed degree and dimension by height, we prove that almost every such hypersurface satisfies the Hasse principle provided that the dimension is at least $3$. This proves a conjecture of Poonen and Voloch in every case except for cubic surfaces.

Seshadri constants and K-stability of Fano manifolds

We give a lower bound of the δ-invariants of ample line bundles in terms of Seshadri constants. As applications, we prove the uniform K-stability of infinitely many families of Fano hypersurfaces of arbitrarily large index, as well as the uniform K-stability of most families of smooth Fano threefolds of Picard number one.

Interpolation of curves on Fano hypersurfaces

On a general hypersurface of degree [Formula: see text] in [Formula: see text] or [Formula: see text] itself, we prove the existence of curves of any genus and high enough degree depending on the genus passing through the expected number [Formula: see text] of general points or incident to a general collection of subvarieties of suitable codimensions. In some cases, we also show that the family of curves through [Formula: see text] fixed points has general moduli as family of [Formula: see text]-pointed curves. These results imply positivity of certain intersection numbers on Kontsevich spaces of stable maps. An arithmetical Appendix A by Chang describes the set of numerical characters ([Formula: see text], curve degree, genus) to which our results apply.

High-pliability Fano hypersurfaces

We show that five of Reid’s Fano 3-fold hyperurfaces containing at least one compound Du Val singularity of type $$cA_n$$ have pliability at least two. The two elements of the pliability set are the singular hypersurface itself, and another non-isomorphic Fano hypersurface of the same degree, embedded in the same weighted projective space, but with different compound Du Val singularities. The birational map between them is the composition of two birational links initiated by blowing up two Type I centres on a codimension 4 Fano 3-fold of $$\mathbb {P}^2 \times \mathbb {P}^2$$ -type having Picard rank 2.

Dimension of the space of conics on a Fano hypersurface

R. Beheshti showed that for a smooth Fano hypersurface X of degree ⩽8 over the complex number field C, the dimension of the space of lines lying in X is equal to the expected dimension. We study the space of conics on X. In this case, if X contains some linear subvariety, then the dimension of the space can be larger than the expected dimension. In this paper, we show that for a smooth Fano hypersurface X of degree ⩽6 over C, and for an irreducible component R of the space of conics lying in X, if the 2-plane spanned by a general conic of R is not contained in X, then the dimension of R is equal to the expected dimension.

Birational rigidity and K-stability of Fano hypersurfaces with ordinary double points

Extending previous results, we prove that for $$n \ge 5$$ all hypersurfaces of degree $$n+1$$ in $${{\mathbb {P}}}^{n+1}$$ with isolated ordinary double points are birational superrigid and K-stable, hence admit a weak Kähler–Einstein metric.

Balanced Curves and Minimal Rational Connectedness on Fano Hypersurfaces

Abstract On a general hypersurface $X$ of degree $n$ (resp. $<n$) in ${\mathbb {P}}^n$ and any $e\geq n-1$ (resp. arbitrarily large $e$), we construct families of rational curves of degree $e$ going through the maximal number of general points. This solves in many cases the problem of rational curve interpolation on $X$.

Asymptotics of the Banana Feynman amplitudes at the large complex structure limit

Recently Bonisch-Fischbach-Klemm-Nega-Safari discovered, via numerical computation, that the leading asymptotics of the l-loop Banana Feynman amplitude at the large complex structure limit can be described by the Gamma class of a degree (1,...,1) Fano hypersurface F in (P^1)^{l+1}. We confirm this observation by using a Gamma-conjecture type result for F.

K-stability of Fano varieties via admissible flags

Abstract We develop a general approach to prove K-stability of Fano varieties. The new theory is used to (a) prove the existence of Kähler-Einstein metrics on all smooth Fano hypersurfaces of Fano index two, (b) compute the stability thresholds for hypersurfaces at generalised Eckardt points and for cubic surfaces at all points, and (c) provide a new algebraic proof of Tian’s criterion for K-stability, amongst other applications.

Restrictions on rational surfaces lying in very general hypersurfaces

AbstractWe study rational surfaces on very general Fano hypersurfaces in$\mathbb {P}^n$, with an eye toward unirationality. We prove that given any fixed family of rational surfaces, a very general hypersurface of degreedsufficiently close tonandnsufficiently large will admit no maps from surfaces in that family. In particular, this shows that for such hypersurfaces, any rational curve in the space of rational curves must meet the boundary. We also prove that for any fixed ratio$\alpha $, a very general hypersurface in$\mathbb {P}^n$of degreedsufficiently close tonwill admit no generically finite maps from a surface satisfying$H^2 \geq \alpha HK$, whereHis the pullback of the hyperplane class from$\mathbb {P}^n$andKis the canonical bundle on the surface.

Stable rationality of index one Fano hypersurfaces containing a linear space

We prove that a very general complex hypersurface of degree \(n+1\) in \(\mathbb {P}^{\,n+1}\) containing an r-plane with multiplicity m is not stably rational under some mild assumptions for \(n \geqslant 3\), \(m, r > 0\). We also prove the failure of stable rationality of a very general hypersurface of degree \(n+1\) in \(\mathbb {P}^{\,n+1}\) admitting several isolated ordinary double points.

Height of rational points on random Fano hypersurfaces

We investigate in a statistical fashion the smallest height of a rational point on a Fano hypersurface defined over the field of rational numbers. Along the way, we establish an average version of Manin’s conjecture about the number of rational points of bounded height on Fano varieties for the complete family of Fano hypersurfaces of fixed degree and dimension.

Torsion orders of Fano hypersurfaces

We find new lower bounds on the torsion orders of very general Fano hypersurfaces over (uncountable) fields of arbitrary characteristic. Our results imply that unirational parametrizations of most Fano hypersurfaces need to have enormously large degree. Our results also hold in characteristic two, where they solve the rationality problem for hypersurfaces under a logarithmic degree bound, thereby extending a previous result of the author from characteristic different from two to arbitrary characteristic.

Low-degree rational curves on hypersurfaces in projective spaces and their fan degenerations

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.

Variation of stable birational types in positive characteristic

Let k be an uncountable algebraically closed field and let Y be a smooth projective k-variety which does not admit a decomposition of the diagonal. We prove that Y is not stably birational to a very general hypersurface of any given degree and dimension. We use this to study the variation of the stable birational types of Fano hypersurfaces over fields of arbitrary characteristic. This had been initiated by Shinder, whose method works in characteristic zero.

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, the invariant ‘the minimal degree of a dominant rational map to a ruled variety’ can only drop on special fibers. As a consequence, we show that for certain low-dimensional families of varieties, the degree of irrationality also behaves well under specialization.