Smooth Threefolds Research Articles

Surface counterexamples to the Eisenbud-Goto conjecture

It is well known that the Eisenbud-Goto regularity conjecture is true for arithmetically Cohen-Macaulay varieties, projective curves, smooth surfaces, smooth threefolds in P 5 \mathbb {P}^5 , and toric varieties of codimension two. After J. McCullough and I. Peeva constructed counterexamples in 2018, it has been an interesting question to find the categories such that the Eisenbud-Goto conjecture holds. So far, surface counterexamples have not been found while counterexamples of any dimension greater or equal to 3 are known. In this paper, we construct counterexamples to the Eisenbud-Goto conjecture for projective surfaces in P 4 \mathbb {P}^4 and investigate projective invariants, cohomological properties, and geometric properties. The counterexamples are constructed via binomial rational maps between projective spaces.

Non-density of Points of Small Arithmetic Degrees

Given a surjective endomorphism $f: X \to X$ on a projective variety over a number field, one can define the arithmetic degree $\alpha_f(x)$ of $f$ at a point $x$ in $X$. The Kawaguchi - Silverman Conjecture (KSC) predicts that any forward $f$-orbit of a point $x$ in $X$ at which the arithmetic degree $\alpha_f(x)$ is strictly smaller than the first dynamical degree $\delta_f$ of $f$ is not Zariski dense. We extend the KSC to sAND (= small Arithmetic Non-Density) Conjecture that the locus $Z_f(d)$ of all points of small arithmetic degree is not Zariski dense, and verify this sAND Conjecture for endomorphisms on projective varieties including surfaces, HyperK\"ahler varieties, abelian varieties, Mori dream spaces, simply connected smooth varieties admitting int-amplified endomorphisms, smooth threefolds admitting int-amplified endomorphisms, and some fibre spaces. We show the equivalence of the sAND Conjecture and another conjecture on the periodic subvarieties of small dynamical degree; we also show the close relations between the sAND Conjecture and the Uniform Boundedness Conjecture of Morton and Silverman on endomorphisms of projective spaces and another long standing conjecture on Uniform Boundedness of torsion points in abelian varieties.

Prelog Chow groups of self-products of degenerations of cubic threefolds

It is unknown whether smooth cubic threefolds have an (integral Chow-theoretic) decomposition of the diagonal, or whether they are stably rational or not in general. As a first step towards making progress on these questions, we compute the (saturated numerical) prelog Chow group of the self-product of a certain degeneration of cubic threefolds.

Examples of Surfaces with Canonical Map of Maximal Degree

It was shown by Beauville that if the canonical map $\varphi_{|K_M|}$ of a complex smooth projective surface $M$ is generically finite, then $\operatorname{deg}(\varphi_{|K_M|}) \leq 36$. The first example of a surface with canonical degree $36$ was found by the second author. In this article, we show that for any surface which is a degree four Galois étale cover of a fake projective plane $X$ with the largest possible automorphism group $\operatorname{Aut}(X) = C_7:C_3$ (the unique non-abelian group of order $21$), the base locus of the canonical map is finite, and we verify that $35$ of these surfaces have maximal canonical degree $36$. We also classify all smooth degree four Galois étale covers of fake projective planes, which give possible candidates for surfaces of canonical degree $36$. Finally, we also confirm in this paper the optimal upper bound of the canonical degree of smooth threefolds of general type with sufficiently large geometric genus, related to earlier work of Hacon and Cai.

On equivalent conjectures for minimal log discrepancies on smooth threefolds

On smooth varieties, the ACC for minimal log discrepancies is equivalent to the boundedness of the log discrepancy of some divisor which computes the minimal log discrepancy. In dimension three, we reduce it to the case when the boundary is the product of a canonical part and the maximal ideal to some power. We prove the reduced assertion when the log canonical threshold of the maximal ideal is either at most one-half or at least one.

Conic bundle fourfolds with nontrivial unramified Brauer group

We derive a formula for the unramified Brauer group of a general class of rationally connected fourfolds birational to conic bundles over smooth threefolds. We produce new examples of conic bundles over P^3 where this formula applies and which have nontrivial unramified Brauer group. The construction uses the theory of contact surfaces and, at least implicitly, matrix factorizations and symmetric arithmetic Cohen--Macaulay sheaves, as well as the geometry of special arrangements of rational curves in P^2. We also prove the existence of universally CH_0-trivial resolutions for the general class of conic bundle fourfolds we consider. Using the degeneration method, we thus produce new families of rationally connected fourfolds whose very general member is not stably rational.

Some constraints on positive entropy automorphisms of smooth threefolds

Suppose that $X$ is a smooth, projective threefold over $\mathbb C$ and that $\phi : X \to X$ is an automorphism of positive entropy. We show that one of the following must hold, after replacing $\phi$ by an iterate: i) the canonical class of $X$ is numerically trivial; ii) $\phi$ is imprimitive; iii) $\phi$ is not dynamically minimal. As a consequence, we show that if a smooth threefold $M$ does not admit a primitive automorphism of positive entropy, then no variety constructed by a sequence of smooth blow-ups of $M$ can admit a primitive automorphism of positive entropy. In explaining why the method does not apply to threefolds with terminal singularities, we exhibit a non-uniruled, terminal threefold $X$ with infinitely many $K_X$-negative extremal rays on $NE(X)$.

Examples of surfaces with canonical map of maximal degree

It was shown by Beauville that if the canonical map $\varphi_{|K_M|}$ of a complex smooth projective surface $M$ is generically finite, then $\operatorname{deg}(\varphi_{|K_M|}) \leq 36$. The first example of a surface with canonical degree $36$ was found by the second author. In this article, we show that for any surface which is a degree four Galois etale cover of a fake projective plane $X$ with the largest possible automorphism group $\operatorname{Aut}(X) = C_7:C_3$ (the unique non-abelian group of order $21$), the base locus of the canonical map is finite, and we verify that $35$ of these surfaces have maximal canonical degree $36$. We also classify all smooth degree four Galois etale covers of fake projective planes, which give possible candidates for surfaces of canonical degree $36$. Finally, we also confirm in this paper the optimal upper bound of the canonical degree of smooth threefolds of general type with sufficiently large geometric genus, related to earlier work of Hacon and Cai.

Examples of surfaces with canonical map of maximal degree

It was shown by Beauville that if the canonical map $\varphi_{|K_M|}$ of a complex smooth projective surface $M$ is generically finite, then $\operatorname{deg}(\varphi_{|K_M|}) \leq 36$. The first example of a surface with canonical degree $36$ was found by the second author. In this article, we show that for any surface which is a degree four Galois etale cover of a fake projective plane $X$ with the largest possible automorphism group $\operatorname{Aut}(X) = C_7:C_3$ (the unique non-abelian group of order $21$), the base locus of the canonical map is finite, and we verify that $35$ of these surfaces have maximal canonical degree $36$. We also classify all smooth degree four Galois etale covers of fake projective planes, which give possible candidates for surfaces of canonical degree $36$. Finally, we also confirm in this paper the optimal upper bound of the canonical degree of smooth threefolds of general type with sufficiently large geometric genus, related to earlier work of Hacon and Cai.

Smooth contractible threefolds with hyperbolic $$\mathbb {G}_{m}$$ G m -actions via polyhedral divisors

The aim of this note is to give an alternative proof of the Theorem 4.1 of Koras and Russell (J Algebr Geom 6(4): 671–695, 1997), that is, a characterization of smooth contractible affine varieties endowed with a hyperbolic action of the group $$\mathbb {G}_{m}\simeq \mathbb {C}^{\text {*}}$$ , using the language of polyhedral divisors developed in Altmann and Hausen (Math Ann 334:557–607, 2006) as generalization of $$\mathbb {Q}$$ -divisors.

McKay Correspondence and New Calabi–Yau Threefolds

In this note, we consider crepant resolutions of the quotient varieties of smooth quintic threefolds by Gorenstein group actions. We compute their Hodge numbers via McKay correspondence. In this way, we find some new pairs $(h^{1,1},\;h^{2,1})$ of Hodge numbers of Calabi-Yau threefolds.

Semi-continuity of stability for sheaves and variation of Gieseker moduli spaces

Abstract We investigate a semi-continuity property for stability conditions for sheaves that is important for the problem of variation of the moduli spaces as the stability condition changes. We place this in the context of a notion of stability previously considered by the authors, called multi-Gieseker-stability, that generalises the classical notion of Gieseker-stability to allow for several polarisations. As such we are able to prove that on smooth threefolds certain moduli spaces of Gieseker-stable sheaves are related by a finite number of Thaddeus-flips (that is flips arising for Variation of Geometric Invariant Theory) whose intermediate spaces are themselves moduli spaces of sheaves.

Threefolds with one apparent double point

The number of apparent double points of an irreducible projective variety X of dimension n in $$\mathbb {P}^{2n+1}$$ is the number of secant lines to X passing through a general point of $$\mathbb {P}^{2n+1}$$ . This classical notion dates back to Severi. Smooth threefolds having one apparent double point have been classified in 2004 by Ciliberto, Mella and Russo. This paper presents examples and a partial classification of singular threefolds having one apparent double point.

Non-commutative width and Gopakumar–Vafa invariants

We show that the non-commutative widths for flopping curves on smooth three-folds introduced by Donovan–Wemyss are described by Katz’s genus zero Gopakumar–Vafa invariants.

Rank 2 vector bundles over $${\mathbb {P}}^{2}(\mathbb {C})$$ P 2 ( C ) whose sections vanish on points in general position

In this paper we address the classification of rank 2 vector bundles \(E,\) over \({\mathbb {P}}^{2}(\mathbb {C}),\) such that the zero locus of a general section of \(E\) is a set of distinct points in general position. We also discuss the existence of smooth threefolds \(X={\mathbb {P}}(E),\) arising from such bundles.

Seshadri constants on smooth threefolds

Abstract We prove that the Seshadri constant of an ample line bundle at a very general point of a smooth projective threefold is larger than 1/2. While falling short of the conjectured lower bound of one, this improves on known results. We systematically exploit new results and ideas related to the variation and complexity of base loci.

New construction of complex manifold via conifold transition

For a generic anti-canonical hypersurface in each smooth toric Fano 4-fold with rank 2 Picard group, we prove there exist three isolated rational curves in it. Moreover, for all these 4-folds except one, the contractions of generic anti-canonical hypersurfaces along the three rational curves can be deformed to smooth threefolds which is diffeomorphic to connected sums of S3 × S3. In this manner, we obtain complex structures with trivial canonical bundles on some connected sums of S3 × S3. This construction is an analogue of that made by Friedman [On threefolds with trivial canonical bundle. In: Complex Geometry and Lie Theory, volume 53 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 1991, 103–134], Lu and Tian [Complex structures on connected sums of S3 × S3. In: Manifolds and Geometry, Pisa, 1993, 284–293] who used only quintics in ℙ4.

Classification of (1, 2)-Grassmann secant defective threefolds

In this paper, we classify all smooth threefolds , for which the Grassmann secant variety G 1,2 ( X ) ⊂ 𝔾(1, N ) (i.e. the closure of the set of lines contained in the span of 3 independent points of X ) has not the expected dimension.

On the quadratic normality and the triple curve of three-dimensional subvarieties of

Abstract A well-known conjecture asserts that smooth threefolds are quadratically normal with the only exception of the Palatini scroll. As a corollary of a more general statement we obtain the following result, which is related to the previous conjecture: If is not quadratically normal, then its triple curve is reducible. Similar results are also given for higher dimensional varieties.

A lower bound on Seshadri constants of hyperplane bundles on threefolds

We give the lower bound on Seshadri constants for the case of very ample line bundles on threefolds. We consider the situation when the Seshadri constant is strictly less than 2 and give a version of Bauer’s theorem (Math Ann 313(3):547–583, 1999, Theorem 2.1) for singular surfaces so we can prove the same result for smooth threefolds.