Obstructions to deforming curves on an Enriques-Fano 3-fold

Dedicated to Professor Shigeru Mukai on the occasion of his 65-th birthday
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Abstract

We study the deformations of a curve C on an Enriques-Fano 3-fold XPn, assuming that C is contained in a smooth hyperplane section SX, that is a smooth Enriques surface in X. We give a sufficient condition for C to be (un)obstructed in X, in terms of half pencils and (2)-curves on S. Let HilbscX denote the Hilbert scheme of smooth connected curves in X. By using the Hilbert-flag scheme of X, we also compute the dimension of HilbscX at [C] and give a sufficient condition for HilbscX to contain a generically non-reduced irreducible component of Mumford type.

Introduction

We work over an algebraically closed field k of characteristic 0. Given a projective scheme X over k, we denote by HilbscX the Hilbert scheme of smooth connected curves in X. Mumford [16] first proved that HilbscP3 contains a generically non-reduced (irreducible) component. In [15], [17], [18] for smooth Fano 3-folds X, HilbscX 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 HilbscX contains a generically non-reduced component whose general member is contained in a smooth hyperplane section SKX in Pic X (=Z[KX]), i.e. a smooth K3 surface S in X.

In this paper, we study the Hilbert scheme HilbscX 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 KX is not a Cartier divisor but numerically equivalent to the hyperplane section S|OX(1)|. The number g:=(KX)3/2+1 is called the genus of X, and it is known that for every X we have g17 (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 |2E| is base-point-free and defines an elliptic fibration on S. (Such a divisor E is called a half pencil on S.) Let NE/X denote the normal bundle of E in X and let NE/X(E) be defined by NE/X(E):=NE/XENE/S. The following is our main theorem.

Theorem 1.1

Let X be an Enriques-Fano 3-fold of genus g, and SX a smooth hyperplane section, i.e. an Enriques surface S in X. If there exists a half pencil E on S of degree e:=(KX.E)2 such that H1(E,NE/X(E))=0, then HilbscX contains a generically non-reduced component W of dimension 2g+2e1 whose general member C satisfies:

  • (1)

    C is contained in an Enriques surface S in X, and

  • (2)

    C is linearly equivalent to KX|S+2E in S for some half pencil E on S.

In Example 4.2 we give a few examples of Enriques-Fano 3-folds X (of genus g=6,9,13) satisfying the assumption of Theorem 1.1. For these X, there exist a smooth Fano 3-fold Y and a K3 surface MY which double cover X and S, respectively. We use the geometry of elliptic fibrations on M to show the existence of the desired half pencil E on S. It might be notable that for these X we have
  • (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 HilbscY of double dimension as HilbscX (i.e. dimV=2dimW for the component WHilbscX), whose general member is contained in a K3 surface M in Y, but M is not general in |KY| (cf. Remark 4.5).

One can compare Theorem 1.1 with Proposition 4.4, which gives a sufficient condition for the Hilbert scheme of a smooth Fano 3-fold to have a generically non-reduced component. Theorem 1.1 is obtained as an application of Theorem 1.2, which enables us to compute the dimension of HilbscX at [C] and determines the (un)obstructedness of C in X for curves C contained in S.

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 H1(S,OS(C))=0. We define a divisor D on S by D:=C+KX|S.

  • (1)

    If H1(S,D)=0, then C is unobstructed in X.

  • (2)

    If there exists a half-pencil E on S such that D2mE or DKS+(2m+1)E for an integer m1, then we have h1(S,D)=m. If moreover H1(E,NE/X(E))=0, then C is obstructed in X.

  • (3)

    If D0, D20 and there exists a (2)-curve E on S such that E.D=2 and H1(S,D3E)=0, then we have h1(S,D)=1. If moreover the π-map πE/S(E) (cf. (2.7)) for (E,S) is not surjective, then C is obstructed in X.

In (1), (2) and (3), if we assume furthermore that h0(S,KSD)=0 and m=1 (for (2)), then HilbscX is of dimension g+g(C)1 at [C], where g(C) denote the (arithmetic) genus of C.

If H1(S,OS(C))=0, then the Hilbert-flag scheme HFscX of X is nonsingular at (C,S) of expected dimension g+g(C)1 (cf. Lemma 2.9). If moreover H1(S,D)=0, then the first projection HFscXHilbscX, (C,S)[C] is smooth at (C,S), and thus Theorem 1.2 (1) follows from a property of smooth morphisms. We partially prove that C is obstructed in X if H1(S,D)0 by using half pencils and (2)-curves on S together with a result in [18]. See [18] for a result on the (un)obstructedness of curves lying on a K3 surface in a smooth Fano 3-fold.

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 Hi(S,OS)=0 for i=1,2 and 2KS0. Every Enriques surface S is isomorphic to the quotient M/θ 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 ψ:SP1, 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 g(C).

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 H1(X,N(C,S)/X)=0, which implies that the Hilbert-flag scheme HFscX of X is nonsingular at (C,S). Moreover, it follows from (2.4) that there exists an exact sequence where p1 is the tangent map of the first projection pr1:HFscX

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 HilbscY also contains a generically non-reduced component and compare its properties (e.g. the dimension of the component) with that of HilbscX (cf. Remark 4.5). In what

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