Obstructions to deforming curves on an Enriques-Fano 3-fold
Introduction
We work over an algebraically closed field k of characteristic 0. Given a projective scheme X over k, we denote by the Hilbert scheme of smooth connected curves in X. Mumford [16] first proved that contains a generically non-reduced (irreducible) component. In [15], [17], [18] for smooth Fano 3-folds X, has been studied from the viewpoint of generalizations of Mumford's example and more recently it has been proved in [19] that if X is a prime Fano 3-fold then contains a generically non-reduced component whose general member is contained in a smooth hyperplane section in Pic X (), i.e. a smooth K3 surface S in X.
In this paper, we study the Hilbert scheme for Enriques-Fano 3-folds X (see Definition 2.3) and discuss the existence of its generically non-reduced components. The 3-folds X in this class contain a (smooth) Enriques surface S as a hyperplane section, and were originally studied by Fano in his famous paper [8] and the study was followed in many papers, e.g., [4], [5], [2], [21], [20], [12]. It is known that every Enriques-Fano 3-fold X has isolated singularities and is not a Cartier divisor but numerically equivalent to the hyperplane section . The number is called the genus of X, and it is known that for every X we have (cf. [20], [12]). It follows from a general theory that on every Enriques surface S there exists an effective divisor E on S such that is base-point-free and defines an elliptic fibration on S. (Such a divisor E is called a half pencil on S.) Let denote the normal bundle of E in X and let be defined by . The following is our main theorem. Theorem 1.1 Let X be an Enriques-Fano 3-fold of genus g, and a smooth hyperplane section, i.e. an Enriques surface S in X. If there exists a half pencil E on S of degree such that , then contains a generically non-reduced component W of dimension whose general member C satisfies: C is contained in an Enriques surface in X, and C is linearly equivalent to in for some half pencil on .
- (1)
every general member C of the component W is contained in a general hyperplane section S of X (cf. Remark 2.7), and
- (2)
for the smooth Fano cover Y of X, there exists a generically non-reduced component V of of double dimension as (i.e. for the component ), whose general member is contained in a K3 surface M in Y, but M is not general in (cf. Remark 4.5).
Theorem 1.2 Let X be an Enriques-Fano 3-fold of genus g, S a smooth hyperplane section of X and C a smooth connected curve on S satisfying . We define a divisor D on S by . If , then C is unobstructed in X. If there exists a half-pencil E on S such that or for an integer , then we have . If moreover , then C is obstructed in X. If , and there exists a -curve E on S such that and , then we have . If moreover the π-map (cf. (2.7)) for is not surjective, then C is obstructed in X.
In (1), (2) and (3), if we assume furthermore that and (for (2)), then is of dimension at , where denote the (arithmetic) genus of C.
The organization of this paper is as follows. In §2.1 and §2.2 we recall some properties of Enriques surfaces and Enriques-Fano 3-folds, respectively. In §2.3 we recall some known results on Hilbert-flag schemes and obstructions to lifting first order deformations of curves on a 3-fold to second order deformations (i.e. primary obstructions). These results will be used in §3 and §4 to prove Theorem 1.2, Theorem 1.1, respectively.
Acknowledgments I would like to thank Prof. Hiromichi Takagi for his comment, which motivated me to research the topic of this paper. I would like to thank Prof. Shigeru Mukai for letting me know examples of Enriques-Fano 3-folds. This paper was written during my stay as a visiting researcher at the department of mathematics at the University of Oslo (UiO), Norway. I thank UiO for providing the facilities. I thank Prof. Kristian Ranestad, Prof. John Christian Ottem and Prof. Jan Oddvar Kleppe for helpful and inspiring discussions during the stay. Last but not least, I thank the referee for giving helpful comments improving the readability and quality of this paper. This work was supported in part by JSPS KAKENHI Grant Numbers JP17K05210 and JP20K03541.
Section snippets
Enriques surfaces
In this section, we recall some properties of Enriques surfaces. We refer to [7] and [1] for proofs and more general theories on Enriques surfaces. A smooth projective surface S is called an Enriques surface if for and . Every Enriques surface S is isomorphic to the quotient of a smooth K3 surface M by a fixed-point-free involution θ of M. Here M is the canonical cover of S and called the K3 cover of S. It is well known that S admits an elliptic fibration , whose
Deformations of curves on Enriques-Fano 3-folds
In this section, we prove Theorem 1.2. Let X be an Enriques-Fano 3-fold of genus g, S an Enriques surface in X and C a smooth connected curve on S of genus .
Proof of Theorem 1.2 We first show a strategy of the proof, which is very similar to that of [18, Theorem 1.2]. By Lemma 2.9, we have , which implies that the Hilbert-flag scheme of X is nonsingular at . Moreover, it follows from (2.4) that there exists an exact sequence where is the tangent map of the first projection
Non-reduced components of the Hilbert scheme
In this section, we prove Theorem 1.1. We also give some examples of Enriques-Fano 3-folds satisfying the assumption of the theorem (cf. Example 4.2). In our examples, every Enriques-Fano 3-fold X has only terminal cyclic quotient singularities and there exists a smooth Fano 3-fold Y that double covers X. We also prove that also contains a generically non-reduced component and compare its properties (e.g. the dimension of the component) with that of (cf. Remark 4.5). In what
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