Let $X$ be a very general hypersurface of degree $d$ in the projective $(n+1)$-space with $n \ge 3$, and $f: X \to Y$ a non-birational surjective morphism to a normal projective variety $Y$. We first prove that $Y$ is a klt Fano variety if ${\rm deg} \, f \ge C$ for some constant $C = C(n, d)$ depending only on $n$ and $d$. Next we prove an optimal upper bound ${\rm deg} \, f \le {\rm deg} \, X$ provided that $Y$ is factorial, ${\rm deg} \, f$ is prime and ${\rm deg} \, f \ge E(n)$ for some constant $E(n)$ (with $E(n) = n(n+1)$ when $Y$ is smooth). As a corollary, we show that $Y\cong {\bf P}^n$ under some conditions on $Y$ and ${\rm deg} \, f$.
We show that for every morphism f between nonsingular hypersurfaces of dimension at least 3 and of general type in projective space, there is an everywhere defined endomorphism F of projective space that restricts to f. As a corollary, we see that if X,Y are nonsingular hypersurfaces of general type of dimension at least 3 such that there is a nonconstant morphism f from X to Y, then degY divides degX with quotient q, and moreover the endomorphism F of projective space is given by polynomials of degree q.
This paper studies the existence of free and very free curves on the degree 5 Fermat hypersurface in P^5 over a field of characteristic 2. We find that such curves exist in degrees 8 and 9 and not in lower degrees.
We prove analogues of several well-known results concerning rational morphisms between quadrics for the class of so-called quasilinear $p$-hypersurfaces. These hypersurfaces are nowhere smooth over the base field, so many of the geometric methods which have been successfully applied to the study of projective homogeneous varieties over fields cannot be used. We are therefore forced to take an alternative approach, which is partly facilitated by the appearance of several non-traditional features in the study of these objects from an algebraic perspective. Our main results were previously known …
We show that every morphism from a degree 5 hypersurface in 4-dimensional projective space to a nonsingular degree 3 hypersurface in 4-dimensional projective space is necessarily constant. In the process, we also classify morphisms from the projective plane to nonsingular cubic threefolds given by degree 3 polynomials.
We study the geometry of the morphism between moduli spaces of hypersurfaces in $\mathbb P^{n-1}$ that sends a smooth hypersurface of degree $d+1$ to its associated hypersurface of degree $n(d-1)$. As a result, we obtain a compactification of the moduli space of smooth hypersurfaces such that the induced rational map from the standard GIT compactification often contracts the discriminant divisor.
Special fibrations of toric varieties have been used by physicists, e.g. the school of Candelas, to construct dual pairs in the study of Het/F-theory duality. Motivated by this, we investigate in this paper the details of toric morphisms between toric varieties. In particular, a complete toric description of fibers - both generic and non-generic -, image, and the flattening stratification of a toric morphism are given. Two examples are provided to illustrate the discussions. We then turn to the study of the restriction of a toric morphism to a toric hypersurface. The details of this can be und…
We build two embedded resolution procedures of a quasi-ordinary singularity of complex analytic hypersurface, by using toric morphisms which depend only on the characteristic monomials associated to a quasi-ordinary projection of the singularity. This result answers an open problem of Lipman in Equisingularity and simultaneous resolution of singularities, Resolution of Singularities, Progress in Mathematics No. 181, 2000, 485-503. In the first procedure the singularity is embedded as hypersurface. In the second procedure, which is inspired by a work of Goldin and Teissier for plane curves (see…
In this paper we define and study real fibered morphisms. Such morphisms arise in the study of real hyperbolic hypersurfaces in projective space and other hyperbolic varieties. We show that real fibered morphisms are intimately connected to Ulrich sheaves admitting positive definite symmetric bilinear forms.
Let X/S be a quasi-projective morphism over an affine base. We develop in this article a technique for proving the existence of closed subschemes H/S of X/S with various favorable properties. We offer several applications of this technique, including the existence of finite quasi-sections in certain projective morphisms, and the existence of hypersurfaces in X/S containing a given closed subscheme C, and intersecting properly a closed set F. Assume now that the base S is the spectrum of a ring R such that for any finite morphism Z -> S, Pic(Z) is a torsion group. This condition is satisfied if…
We prove that a general determinantal hypersurface of dimension 3 is nodal. Moreover, in terms of Chern classes associated with bundle morphisms, we derive a formula for the intersection homology Euler characteristic of a general determinantal hypersurface.
In this note we study rational curves on degree $p^r+1$ Fermat hypersurface in $\PP^{p^r+1}_k$, where $k$ is an algebraically closed field of characteristic $p$. The key point is that the presence of Frobenius morphism makes the behavior of rational curves to be very different from that of charateristic 0. We show that if there exists $N_0$ such that for all $e\geq N_0$ there is a degree $e$ very free rational curve on $X$, then $N_0> p^r(p^r-1)$.
We use a counting argument and surgery theory to show that if $D$ is a sufficiently general algebraic hypersurface in $\Bbb C^n$, then any local diffeomorphism $F:X \to \Bbb C^n$ of simply connected manifolds which is a $d$-sheeted cover away from $D$ has degree $d=1$ or $d=\infty$ (however all degrees $d > 1$ are possible if $F$ fails to be a local diffeomorphism at even a single point). In particular, any étale morphism $F:X \to \Bbb C^n$ of algebraic varieties which covers away from such a hypersurface $D$ must be birational.
Let ${\mathcal Q}_n^d$ be the vector space of homogeneous forms of degree $d\ge 3$ on ${\mathbb C}^n$, with $n\ge 2$. The object of our study is the map $Φ$, introduced in earlier articles by J. Alper, M. Eastwood and the author, that assigns to every form for which the discriminant $Δ$ does not vanish the so-called associated form lying in the space ${\mathcal Q}_n^{n(d-2)*}$. This map is a morphism from the affine variety $X_n^d:=\{f\in{\mathcal Q}_n^d:Δ(f)\ne 0\}$ to the affine space ${\mathcal Q}_n^{n(d-2)*}$. Letting $p$ be the smallest integer for which the product $Δ^pΦ$ extends to a mo…
In this paper, we extend our result in [3] to hypersurfaces of any smooth projective variety $Y$. Precisely we let $X_0$ be a generic hypersurface of $Y$ and $c_0:\mathbf P^1\to X_0$ be a generic birational morphism to its image, i.e. $c_0\in Hom_{bir}(\mathbf P^1, X_0)$ is generic, such that (1) $dim(X_0)\geq 3$, (2) $H^1( N_{c_0/Y})=0$. Then \begin{equation} H^1(N_{c_0/X_0})=0. \end{equation} As an application we prove that the Clemens' conjecture holds for Calabi-Yau complete intersections of dimension 3.
The decomposition theorem for smooth projective morphisms $π:\mathcal{X}\rightarrow B$ says that $Rπ_*\mathbb{Q}$ decomposes as $\oplus R^iπ_*\mathbb{Q}[-i]$. We describe simple examples where it is not possible to have such a decomposition compatible with cup-product, even after restriction to Zariski dense open sets of $B$. We prove however that this is always possible for families of $K3$ surfaces (after shrinking the base), and show how this result relates to a result by Beauville and the author on the Chow ring of $K3$ surfaces $S$. We give two proofs of this result, the second one involv…
We 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 degree $d$ sufficiently close to $n$ and $n$ sufficiently 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 $α$, a very general hypersurface in $\mathbb{P}^n$ of degree $d$ sufficiently close to $n$ will admit no maps fr…
In this article we study Brill-Noether loci of moduli space of stable bundles over smooth surfaces. We define Petri map as an analogy with the case of curves. We show the non-emptiness of certain Brill-Noether loci over very general quintic hypersurface in $\mathbb{P}^3$, and use the Petri map to produce components of expected dimension.
We introduce spectral Hirzebruch-Milnor classes for singular hypersurfaces. These can be identified with Steenbrink spectra in the isolated singularity case, and may be viewed as their global analogues in general. Their definition uses vanishing cycles of mixed Hodge modules and the Todd class transformation. These are compatible with the pushforward by proper morphisms, and the classes can be calculated by using resolutions of singularities. Formulas for Hirzebruch-Milnor classes of projective hypersurfaces in terms of these classes are given in the case where the multiplicity of a generic hy…
We relate the geometry of curves to the notion of hyperbolicity in real algebraic geometry. A hyperbolic variety is a real algebraic variety that (in particular) admits a real fibered morphism to a projective space whose dimension is equal to the dimension of the variety. We study hyperbolic varieties with a special interest in the case of hypersurfaces that admit a real algebraic ruling. The central part of the paper is concerned with secant varieties of real algebraic curves where the real locus has the maximal number of connected components, which is determined by the genus of the curve. Fo…
We study the algebraic hyperbolicity of very general hypersurfaces in $\mathbb{P}^m \times \mathbb{P}^n$ by using three techniques that build on past work by Ein, Voisin, Pacienza, Coskun and Riedl, and others. As a result, we completely answer the question of whether or not a very general hypersurface of bidegree $(a,b)$ in $\mathbb{P}^m \times \mathbb{P}^n$ is algebraically hyperbolic, except in $\mathbb{P}^3 \times \mathbb{P}^1$ for the bidegrees $(a,b)= (7,3), (6,3)$ and $(5,b)$ with $b\geq 3.$ As another application of these techniques, we improve the known result that very general hypers…
We prove results concerning the behavior of Hodge ideals under restriction to hypersurfaces or fibers of morphisms, and addition. The main tool is the description of restriction functors for mixed Hodge modules by means of the $V$-filtration.
In this paper we further develop a Grassmannian technique used to prove results about very general hypersurfaces and present three applications. First, we provide a short proof of the Kobayashi Conjecture given previous results on the Green-Griffiths-Lang Conjecture. Second, we characterize the dimension of the space of Chow-equivalent points on a very general hypersurface, proving the remaining cases of a conjecture of Chen, Lewis and Sheng and providing a short, alternate proof for many of the already known cases. Finally, we relate Seshadri constants of very general points to Seshadri const…
Let $C$ be a smooth projective curve of genus $g\geq 4$ over the complex numbers and ${\cal SU}^s_C(r,d)$ be the moduli space of stable vector bundles of rank $r$ with a fixed determinant of degree $d$. In the projectivized cotangent space at a general point $E$ of ${\cal SU}^s_C(r,d)$, there exists a distinguished hypersurface ${\cal S}_E$ consisting of cotangent vectors with singular spectral curves. In the projectivized tangent space at $E$, there exists a distinguished subvariety ${\cal C}_E$ consisting of vectors tangent to Hecke curves in ${\cal SU}^s_C(r,d)$ through $E$. Our main result…
For a complex irreducible projective variety, the volume function and the higher asymptotic cohomological functions have proven to be useful in understanding the positivity of divisors as well as other geometric properties of the variety. In this paper, we study the vanishing properties of these functions on specific hypersurfaces of P^n \times P^n. In particular, we show that very general hypersurfaces of bidegree (k,k) obey a very strong vanishing property, which we define as asymptotic purity: at most one asymptotic cohomological function is nonzero for each divisor. This provides evidence …
We study the Chow motive (with rational coefficients) of a hypersurface X in the projective space by using the variety F(X) of l-dimensional planes contained in X. If the degree of X is sufficiently small we show that the primitive part of the motive of X is the tensor product of a direct summand in the motive of a suitable complete intersection in F(X) and the l-th twist Q(-l) of the Lefschetz motive.
We show that a wide class of hypersurfaces in all dimensions are not stably rational. Namely, for all d at least about 2n/3, a very general complex hypersurface of degree d in P^{n+1} is not stably rational. The statement generalizes Colliot-Thelene and Pirutka's theorem that very general quartic 3-folds are not stably rational. The result covers all the degrees in which Kollar proved that a very general hypersurface is non-rational, and a bit more. For example, very general quartic 4-folds are not stably rational, whereas it was not even known whether these varieties are rational.
We determine the rationality of very general quasismooth Fano 3-fold weighted hypersurfaces completely and determine the stable rationality of them except for cubic 3-folds. More precisely we prove that (i) very general Fano 3-fold weighted hypersurfaces of index 1 or 2 are not stably rational except possibly for the cubic threefolds, (ii) among the 27 families of Fano 3-fold weighted hypersurfaces of index greater than 2, very general members of specific 7 families are not stably rational and the remaining 20 families consists of rational varieties.
For any smooth complex projective variety X and smooth very ample hypersurface Y in X, we develop the technique of genus zero relative Gromov-Witten invariants of Y in X in algebro-geometric terms. We prove an equality of cycles in the Chow groups of the moduli spaces of relative stable maps that relates these relative invariants to the Gromov-Witten invariants of X and Y. Given the Gromov-Witten invariants of X, we show that these relations are sufficient to compute all relative invariants, as well as all genus zero Gromov-Witten invariants of Y whose homology and cohomology classes are induc…
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 for $n \ge 3$, $m, r > 0$ and $n \ge m+r$. We also investigate failure of stable rationality of a very general hypersurface of degree $n+1$ in $\mathbb{P}^{n+1}$ admitting several isolated ordinary double points.
We show that the degrees of rational endomorphisms of very general complex Fano and Calabi-Yau hypersurfaces satisfy certain congruence conditions by specializing to characteristic p. As a corollary we show that very general n-dimensional hypersurfaces of degree $d\ge 5\lceil (n+3)/6\rceil$ are not birational to elliptic fibrations. A key part of the argument is to resolve singularities of general p-cyclic covers in mixed characteristic p.
We show that very general hypersurfaces in odd-dimensional simplicial projective toric varieties verifying a certain combinatorial property satisfy the Hodge conjecture (these include projective spaces). This gives a connection between the Oda conjecture and Hodge conjecture. We also give an explicit criterion which depends on the degree for very general hypersurfaces for the combinatorial condition to be verified.
In this article, we study the infinitemisal invariant of the relative higher Abel Jacobi map of a smooth open morphism. We give a generalization of a theorem of Voisin to open varieties and higher Chow groups and as a corollary a non vanishing criterion for the higher Abel Jacobi map of a general open smooth hypersurface section of high degree of a smooth projective variety Y. On the other side, using Nori connectness theorem, the image of the primitive part of the higher Abel Jacobi map of a general open smooth hypersurface section of high degree of a smooth projective variety Y is generated …
Using Voisin's method we prove that a very general hypersurface of degree at least 4 in complex projective space of dimension 6, 7, 8 or 9 is not stably rational and so, in particular, not rational. We obtain the same conclusion for the double covering of projective space of dimension 6, 7, 8 or 9, branched along a very general quartic hypersurface. On the other hand, such double coverings as well as general quartic hypersurfaces of dimension at least 5 are known to be unirational.
We study a kind of modification of an affine domain which produces another affine domain. First appeared in passing in the basic paper of O. Zariski (1942), it was further considered by E.D. Davis (1967). The first named author applied its geometric counterpart to construct contractible smooth affine varieties non-isomorphic to Euclidean spaces. Here we provide certain conditions which guarantee preservation of the topology under a modification. As an application, we show that the group of biregular automorphisms of the affine hypersurface $X \subset C^{k+2}$ given by the equation $uv=p(x_1,..…
We prove a conjecture of Voisin that no two distinct points on a very general hypersurface of degree $2n$ in ${\mathbb P}^n$ are rationally equivalent.
We develop a technique that allows us to prove results about subvarieties of general type hypersurfaces. As an application, we use a result of Clemens and Ran to prove that a very general hypersurface of degree (3n+1)/2 \leq d \leq 2n-3 contains lines but no other rational curves.
We make a very detailed analysis of the numerical properties of effective divisors whose support is contained in the exceptional locus of a birational morphism of smooth projective surfaces. As an application we extend Miyaoka's inequality on the number of canonical singularities on a projective normal surface with non-negative Kodaira dimension to the non minimal case, obtaining a slightly better result than known extensions by Megyesi and Langer.
We prove the failure of stable rationality for many smooth well formed weighted hypersurfaces of dimension at least 3. It is in particular proved that a very general smooth well formed Fano weighted hypersurface of index one is not stably rational.
This paper presents two approaches to reducing problems on $2$-cycles on a smooth cubic hypersurface $X$ over an algebraically closed field of characteristic $\neq 2$, to problems on $1$-cycles on its variety of lines $F(X)$. The first one relies on bitangent lines of $X$ and Tsen-Lang theorem. It allows to prove that $\mathrm{CH}_2(X)$ is generated, via the action of the universal $\mathbb P^1$-bundle over $F(X)$, by $\mathrm{CH}_1(F(X))$. When the characteristic of the base field is $0$, we use that result to prove that if $dim(X)\geq 7$, then $\mathrm{CH}_2(X)$ is generated by classes of pl…
Let $X$, $Y$ be Fano threefolds of Picard number one and such that the ample generators of Picard groups are very ample. Let $X$ be of index one and $Y$ be of index two. It is shown that the only morphisms from $X$ to $Y$ are double coverings. In fact nearly the whole paper is the analysis of the case where $Y$ is the linear section of the Grassmannian G(1,4), since the other cases were more or less solved in another article. This remaining case is treated with the help of Debarre's connectedness theorem for inverse images of Schubert cycles.
Suppose M is a complex manifold of dimension $n+1$ and K is a hypersurface in M. By Poincaré duality we define a residue morphism $res:H^{k+1}(M\setminus K)\longrightarrow H_{2n-k}(K)$ which generalizes the classical Leray residue morphism to cohomology for smooth K. We assume that K has isolated quasihomogeneous singularities. Suppose $ω$ is a holomorphic form of the type $(n+1,0)$ with the first order pole on K. The purpose of this note is to give a short, self contained proof of a criterion which tells us when the residue of $ω$ lifts to the intersection homology of K.
An enumerative problem on a variety $V$ is usually solved by reduction to intersection theory in the cohomology of a compactification of $V$. However, if the problem is invariant under a "nice" group action on $V$ (so that $V$ is spherical), then many authors suggested a better home for intersection theory: the direct limit of the cohomology rings of all equivariant compactifications of $V$. We call this limit the affine cohomology of $V$ and construct affine characteristic classes of subvarieties of a complex torus, taking values in the affine cohomology of the torus. This allows us to make t…
Let $Y$ be a smooth hypersurface in a projective irreducible holomorphic symplectic manifold $X$ of dimension $2n$. The characteristic foliation $F$ is the kernel of the symplectic form restricted to $Y$. Assume that $X$ is equipped with a Lagrangian fibration $π:X\to B$ and $Y=π^{-1}D$, where $D$ is a hypersurface in $B$. It is easy to see that the leaves of $F$ are contained in the fibers of $π$. We prove that a very general leaf is Zariski dense in a fiber of $π$.
Given a hypersurface in the complex projective $n$-space we prove several known formulas for the degree of its polar map by purely algebro-geometric methods. Furthermore, we give formulas for the degree of its polar map in terms of the degrees of the polar maps of its components. As an application, we classify the plane curves with polar map of low degree, including a very simple proof of I. Dolgachev's classification of homaloidal plane curves.
In this article we give a sufficient condition for a morphism $φ$ from a smooth variety $X$ to projective space, finite onto a smooth image, to be deformed to an embedding. This result puts some theorems on deformation of morphisms of curves and surfaces such as $K3$ and general type, obtained by ad hoc methods, in a new, more conceptual light. One of the main interests of our result is to apply it to the construction of smooth varieties in projective space with given invariants. We illustrate this by using our result to construct canonically embedded surfaces with $c_1^2=3p_g-7$ and derive so…
