We prove an effective bound for the degree of a smooth divisor of a hypersurface of P^n, n>4 (projective space over an algebraically closed field of characteristic zero). Our result follows from a strong (since the degree of the divisor is not involved) generalization of the "Speciality theorem" of Gruson-Peskine, which we prove to hold for any smooth, subcanonical, codimension two, projective verieties of dimension at least three.
We find polyhedral divisors corresponding to the torus action of complexity one on affine trinomial hypersurfaces. Explicit computations for particular classes of such hypersurfaces including Pham-Brieskorn surfaces and rational trinomial hypersurfaces are given.
In this note, we extend work of Farkas and Rimányi on applying quadric rank loci to finding divisors of small slope on the moduli space of curves by instead considering all divisorial conditions on the hypersurfaces of a fixed degree containing a projective curve. This gives rise to a large family of virtual divisors on $\overline{\mathcal{M}_g}$. We determine explicitly which of these divisors are candidate counterexamples to the Slope Conjecture. The potential counterexamples exist on $\overline{\mathcal{M}_g}$, where the set of possible values of $g\in \{1,\ldots,N\}$ has density $Ω(\log(N)…
The Chern class of the sheaf of logarithmic derivations along a simple normal crossing divisor equals the Chern-Schwartz-MacPherson class of the complement of the divisor. We extend this equality to more general divisors, which are locally analytically isomorphic to free hyperplane arrangements.
We introduce a dual logarithmic residue map for hypersurface singularities and use it to answer a question of Kyoji Saito. Our result extends a theorem of Lê and Saito by an algebraic characterization of hypersurfaces that are normal crossing in codimension one. For free divisors, we relate the latter condition to other natural conditions involving the Jacobian ideal and the normalization. This leads to an algebraic characterization of normal crossing divisors. As a side result, describe all free divisors with Gorenstein singular locus.
Totally invariant divisors of endomorphisms of the projective space are expected to be always unions of linear spaces. Using logarithmic differentials we establish a lower bound for the degree of the non-normal locus of a totally invariant divisor. As a consequence we prove the linearity of totally invariant divisors for $\mathbb P^3$.
We show that a smooth divisor in a projective space can be reconstructed from the isomorphism class of the sheaf of logarithmic vector fields along it if and only if its defining equation is of Sebastiani-Thom type.
For a hypersurface in a projective space, we consider the set of pairs of a point and a line in the projective space such that the line intersects the hypersurface at the point with a fixed multiplicity. We prove that this set of pairs forms a smooth variety for a general hypersurface.
In this paper, we study the structure of projective space bundles whose relative anti-canonical line bundle is nef. As an application, we get a characterization of abelian varieties up to finite etale covering.
We compute the divisor class group and the Picard group of projective varieties with Hibi rings as homogeneous coordinate rings. These varieties are precisely the toric varieties associated to order polytopes. We use tools from the theory of toric varieties to get a description of the two groups which only depends on combinatorial properties of the underlying poset.
Let $X$ be a projective manifold of dimension $n$. Suppose that $T_X$ contains an ample subsheaf. We show that $X$ is isomorphic to $\mathbb{P}^n$. As an application, we derive the classification of projective manifolds containing a $\mathbb{P}^r$-bundle as an ample divisor by the recent work of D.~Litt.
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.
In this paper, we discuss classical derivation of the residue integral representation of the d=1 rational Gromov-Witten invariants of projective hypersurfaces that followed from localization technique.
In this paper we present some properties for projective hypersurfaces, smooth and singular, to be criteria for identification. To make the decision with these criteria, we have included procedures written in Singular language.
Given a web (multi-foliation) and a linear system on a projective surface we construct divisors cutting out the locus where some element of the linear system has abnormal contact with the leaf of the web. We apply these ideas to reobtain a classical result by Salmon on the number of lines on a projective surface. In a different vein, we investigate the number of lines and of disjoint lines contained in a projective surface and tangent to a contact distribution.
In this paper we classify all the compactifications of affine homology $3$-cells into the blow-ups of the projective $3$-space along smooth curves such that the log canonical divisors are linearly trivial. As a result, we prove that each embedded affine $3$-fold is isomorphic to the affine $3$-space except one example.
We prove that a smooth surface, non of general type, in projective four-space, which lies on a quartic hypersurface with isolated singularities has degree at most 27 (in fact we prove a slightly more general result).
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.
This paper continues our researches \cite{DS1, DS2, DS3} by computing some invariants based on Hilbert-Poincaré series associated to Milnor algebras. Our computations are for some of the classical surfaces and 3-folds with different configurations of isolated singularities. As a by-product of a recent result of E. Sernesi, we give examples of classical hypersurfaces which are (or are not) projectively rigid. We also include a Singular program to compute the invariants and to decide if a singular projective hypersurface is nodal and projectively rigid.
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 determine the complete list of anticanonically embedded quasi smooth log Fano 3-folds in weighted projective 4-spaces. This implies that the Reid-Fletcher list of 95 types of anticanonically embedded quasi smooth terminal Fano threefolds in weighted projective 4-spaces is complete. We also prove that there are only finitely many families of quasi smooth Calabi-Yau hypersurfaces in weighted projective spaces of any given dimension.
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 …
Using the data schemes developed by Arrondo-Sols-Speiser, we give a rigorous definition of algebraic differential equations on the complex projective space $P^n$. For an algebraic subvariety $S \subseteq P^n$, we present an explicit formula for the degree of the divisor of solutions of a differential equation on $S$ and give some examples of applications. We extend the technique and result to the real case.
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 classify four-dimensional quasismooth weighted hypersurfaces with small canonical class, and verify a conjecture of Johnson and Kollar on infinite series of quasismooth hypersurfaces with anticanonical hyperplane section in the case of fourfolds. By considering the quotient singularities that arise, we classify those weighted hypersurfaces that are canonical, Calabi-Yau, and Fano fourfolds. We also consider other classes of hypersurfaces, including Fano hypersurfaces of index greater than 1 in dimensions 3 and 4.
We study the birational properties of hypersurfaces in products of projective spaces. In the case of hypersurfaces in P^m x P^n, we describe their nef, movable and effective cones and determine when they are Mori dream spaces. Using these results, we give new simple examples of non-Mori dream spaces and analogues of Mumford's example of a strictly nef line bundle which is not ample.
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.
A general method of computing cohomology groups of the space of nonsingular algebraic hypersurfaces of degree $d$ in $CP^n$ is described. Using this method, rational cohomology groups of such spaces with $n=2, d \le 4$ and $n=3=d$ are calculated.
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.
Let $X$ be a four-dimensional projective variety defined over the field of complex numbers with only terminal singularities. We prove that if the intersection number of the canonical divisor $K$ with every very general curve is positive ($K$ is almost numerically positive) then every very general proper subvariety of $X$ is of general type in the viewpoint of geometric Kodaira dimension. We note that the converse does not hold for simple abelian varieties.
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 study subvarieties of a general projective degree $d$ hypersurface $X_d\subset \mathbf P^n$. Our main theorem, which improves previous results of L. Ein and C. Voisin, implies in particular the following sharp corollary: any subvariety of a general hypersurface $X_{d}\subset {\mathbf P}^n$, for $n\geq 6$ and $d\geq 2n-2$, is of general type.
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…
In response to a question of Reid, we find all anti-canonical Calabi-Yau hypersurfaces $X$ in toric weighted projective bundles over the projective line where the general fiber is a weighted K3 hypersurface. This gives a direct generalization of Reid's discovery of the 95 families of weighted K3 hypersurfaces. We also treat the case where $X$ is fibered over the plane with general fiber a genus one curve in a weighted projective plane.
Let V be a projective hypersurface of fixed degree and dimension which has only isolated singular points. We show that, if the sum of the Milnor numbers at the singular points of V is large, then V cannot have a point of large multiplicity, unless V is a cone. As an application, we give an affirmative answer to a conjecture of Dimca and Papadima.
By an additive action on a hypersurface H in the projective space P^{n+1} we mean an effective action of a commutative unipotent group on P^{n+1} which leaves H invariant and acts on H with an open orbit. Brendan Hassett and Yuri Tschinkel have shown that actions of commutative unipotent groups on projective spaces can be described in terms of local algebras with some additional data. We prove that additive actions on projective hypersurfaces correspond to invariant multilinear symmetric forms on local algebras. It allows us to obtain explicit classification results for non-degenerate quadrics…
For every positive integer $n\in \mathbb{Z}_+$ we define an `Euler polynomial' $\mathscr{E}_n(t)\in \mathbb{Z}[t]$, and observe that for a fixed $n$ all Chern numbers (as well as other numerical invariants) of all smooth hypersurfaces in $\mathbb{P}^n$ may be recovered from the single polynomial $\mathscr{E}_n(t)$. More generally, we show that all Chern classes of hypersurfaces in a smooth variety may be recovered from its top Chern class.
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 investigate the 0-th local cohomology of the Jacobian ring of a homogeneous polynomial defining a projective hypersurface whose singular locus is a 0-dimensional complete intersection.
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.
