The space of twisted cubics

We consider the Cohen-Macaulay compactification of the space of twisted cubics in projective n-space. This compactification is the fine moduli scheme representing the functor of CM-curves with Hilbert polynomial 3t+1. We show that the moduli scheme of CM-curves in projective 3-space is isomorphic to the twisted cubic component of the Hilbert scheme. We also describe the compactification for twisted cubics in n-space.


Introduction
The main purpose of the present article is to provide a natural and functorial compactification of the space of twisted cubics in P 3 . The representing object is a smooth and irreducible projective scheme that is isomorphic to the twisted cubic component of the Hilbert scheme.
A twisted cubic is a smooth rational curve of degree three. The family of such curves in P 3 is twelve dimensional. Finding a compactification of the parameter space with control over the degenerations is a fundamental question. An immediate answer is provided by the Hilbert scheme. In their celebrated work [PS85] Piene and Schlessinger described the Hilbert scheme Hilb 3t+1 An appealing feature of the Hilbert scheme is that it represents a functor, and thereby comes with universal defining properties. It gives a description of the families of curves including their degenerations. Its disadvantage is clear from the description above -it parametrizes also objects that are geometrically quite different from twisted cubics. Kontsevich's moduli space of stable maps M 0,0 (P 3 , 3) is another functorial compactification of the space of twisted cubics [Kon95,CK11]. The representing stack has been a marvelous tool for enumerative problems, but the geometry of this compactification is quite different from the twisted cubic component H.
An interpolation between the Hilbert scheme and the Kontsevich moduli space was introduced by Hønsen [Høn05]. We refer to it as the space of CM-curves. A Cohen-Macaulay scheme X of pure dimension one, together with a finite morphism X −→ P n that generically is an immersion is a CM-curve (see Section 2). The functor of CM-curves in P n with a fixed Hilbert polynomial is represented by a proper algebraic space ( [Høn05], [Hei14]).
In the present article we use the space of CM-curves with Hilbert polynomial 3t + 1 to compactify the space of twisted cubics. The most interesting case is that of curves in P 3 . Our main result (see Theorems 2.16 and 5.3) gives in particular a modular description of the twisted cubic component H:

Fitting ideals
The n th Fitting ideal sheaf of a coherent O X -module E is denoted by Fitt n X (E), or simply Fitt n (E). We refer to [Nor76] or [Eis95] for definitions and their basic properties. We will be particularly interested in the 0 th Fitting ideal, inspired by the discussion by Teissier in [Tei77]. The ideal sheaf Fitt 0 X (E) is obtained from the maximal minors of the matrix in a presentation of the module E.

CM-curves
We recall the notion of CM-curves as introduced by Hønsen [Høn05]. Let P n denote the projective n-space over a fixed algebraically closed field k. A finite morphism i : X −→ P n is a CM-curve if (1) the scheme X is Cohen-Macaulay and is of pure dimension one, (2) the morphism i is, apart from a finite set, an isomorphism onto its schematic image.
The Hilbert polynomial of a CM-curve i : X −→ P n is the Hilbert polynomial of the coherent sheaf i * O X on P n .

Functor of Cohen-Macaulay curves
Let p(t) = at+b be a numerical polynomial. For a scheme S we let CM p(t) P n (S) denote the set of isomorphism classes of pairs (X, i) where i : X −→ P n S = P n × S is a finite morphism of S-schemes, and where X −→ S is flat, and for every geometric point in S the fiber is a CM-curve with Hilbert polynomial p(t) = at + b. Two pairs (X, i) and (X , i ) are isomorphic if there exists an S-isomorphism α : X −→ X commuting with the maps i and i . We have that CM p(t) P n is a contravariant functor from the category of locally Noetherian schemes, over our fixed field k, to Sets. Theorem 2.5 (Hønsen, Heinrich). The functor CM p(t) P n of CM-curves in P n having a fixed Hilbert polynomial p(t) = at + b, is representable by a proper algebraic space.
Theorem 2.5 is proven more generally in [Høn05] for schemes defined over a fixed field of arbitrary characteristic, not necessarily algebraically closed. A different construction and proof is given in [Hei14] where it is shown that the functor is representable by a proper algebraic space of finite type over Spec Z.

Tangent space to CM
With P n over a field k and the polynomial p(t) fixed, we write CM(R) for CM p(t) P n (Spec(R)) for any kalgebra R. Given a pair ξ = (X, i) in CM(k) we define the functor CM ξ [Høn05, §3.4] on the category of local artinian k-algebras by This is in fact a deformation functor.
For a morphism i : X −→ Y Buchweitz [Buc81] considers six deformation functors, one of them being Def X/Y , the functor of deformations of X over Y , where one deforms X and the morphism i, but not the target Y . Its tangent space is T 1 . The functor CM ξ is the functor Def X/P n of deformations of X over P n , the completion of the local ring of the point (X, i) ∈ CM(k) is the versal deformation of X over P n and the tangent space is CM ξ (k[ε]), where ε 2 = 0. In our specific situation we shall compute this tangent space by directly describing infinitesimal deformations of (X, i). The space of twisted cubics 5

Conductor
Let D = i(X) denote the schematic image of a CM-curve i : X −→ P n . We have the short exact sequence of sheaves on D. The target double point scheme [KLU92,p.204] of i : X −→ P n is the closed subscheme N X ⊆ D defined by the ideal sheaf This ideal sheaf equals the conductor of O D in i * O X . By definition the underlying set |N X | of points in D is a finite set, and the open set D \ N X is isomorphic to X \ i −1 (N X ).

Twisted cubics
The easiest example of a space of CM-curves is that of plane curves of degree d, which is a space that coincides with the Hilbert scheme. One of the easiest interesting, non-planar examples is the case of CMcurves with Hilbert polynomial 3t + 1. In what follows we will assume that P n is the projective n-space defined over an algebraically closed field.
Proposition 2.9. Let i : X −→ P n be a CM-curve where i is not a closed immersion. Assume that the Hilbert polynomial of i * O X is 3t + 1. Then the image D = i(X) is a plane cubic curve, and there is exactly one point P ∈ D over which the morphism i fails to be an isomorphism.
Proof. The Hilbert polynomial of the skyscraper sheaf K = i * O X /O D is a positive constant > 0; it is non-zero since by assumption the map i is not a closed immersion. The Hilbert polynomial of D is then p(t) = 3t + 1 − . Hartshorne proved [Har94, Theorem 3.1] that the arithmetic genus p a = 1 − p(0) of the space curve D ⊆ P 3 is bounded above by p a ≤ 1 2 (d − 1)(d − 2), where d is the degree of D, with equality if and only if D is a plane curve. The same proof applies for D ⊂ P n , where n ≥ 3 is arbitrary. It follows that = 1. The support of K is then one point on D. The equality p a = 1 2 (d − 1)(d − 2) implies that the curve D is planar.
If the morphism i is not a closed immersion then by Proposition 2.9 the image of X is a planar cubic D. Tensoring the exact sequence (2.7.1) with O P n (1) gives the exact sequence of sheaves on P n . We have that dim k H 0 (D, O D (1)) = 3 and that H 1 (D, O D (1)) = 0 as D is a planar cubic. It then follows from the exact sequence that H 1 (P n , i * O X (1)) = 0 and that dim k H 0 (P n , i * O X (1)) = 4. The projection formula gives that i * O X (1) = i * (i * O P n (1)), and since the morphism i is finite the result follows.
Proposition 2.11. Let i R : X R −→ P n R be a CM-curve with Hilbert polynomial 3t + 1. Assume that R is a local ring.
(1) We have that H 0 (X R , i * R O P n R (1)) is a free R-module of rank 4. (2) Let us fix a basis of H 0 (X R , i * R O P n R (1)) and let j R : X R −→ P 3 R = Proj(R[y 0 , y 1 , y 2 , y 3 ]) be the induced morphism. Then j R is a closed immersion.
(4) The pairs (X R , i R ) and (j R (X R ), h R ) are equal as points of the CM-functor.
Proof. Let i : X −→ P 3 be the curve over the closed point of Spec(R). By Lemma 2.10 we have that H r (X, i * O P n (1)) = 0 for r > 0. Let f R : X R −→ Spec(R) be the structure map. It follows by cohomology and base change [Gro63,Corollaire 7.9.9] that f R * i * R O P n R (1) is free, and therefore by Lemma 2.10 free of rank 4. A choice of basis (s 0 , s 1 , s 2 , s 3 ) of global sections of j * R O(1) gives a morphism j R : , which can be expressed linearly in the basis (s 0 , s 1 , s 2 , s 3 ). This defines the map h R . To show that the finite morphism j R is a closed immersion it suffices to show that the induced map j : X −→ P 3 over the closed point of Spec(R) is a closed immersion. If j is not a closed immersion then we have by Proposition 2.9 that the image is contained in a plane in P 3 . This contradicts the fact that the global sections s 0 = j * (y 0 ), . . . , s 3 = j * (y 3 ) are linearly independent. It follows from the construction that the pairs (j R (X R ), h R ) and (X R , i R ) are isomorphic.

Tangent space calculations
The purpose of the rest of the present section is to prove that the CM-spaces with Hilbert polynomial 3t + 1 are smooth. Smoothness is proven by showing that the dimension of the tangent space at the most special point equals the dimension of a general, smooth point. Later we will see that the most special situation is a projection of the curve X ⊂ P 3 given by the graded ideal I = (u 2 , uy − x 2 , xu) ⊂ k[x, y, u, w]. Thus, for the moment we focus on this particular ideal. The canonical morphism k[x, y, w] −→ k[x, y, u, w]/I of graded rings determines the finite morphism (2.12.1) i 1 : X −→ P 2 = Proj(k[x, y, w]).
As such we have a CM-curve in P 2 with Hilbert polynomial 3t + 1. We will also view P 2 as a closed subscheme in P 3 given by the homogeneous ideal ( x + a 2 w a 7 y + a 6 w u y + a 1 w u + a 5 x + a 4 y + a 3 w x + a 8 w , and the morphism i R : X R −→ P 3 R = Proj(R[x, y, z, w]) is determined by the linear map i * R (x, y, z, w) = (x, y, b 9 x + b 10 y + b 11 w + b 12 u, w). Proof. We need to describe all deformations of the pair (X, i). By Proposition 2.11 we may assume for a deformation (X R , i R ) over a local ring R that the scheme X R is embedded in some P 3 R and the morphism i R is induced by a linear map. We start with infinitesimal deformations. The embedded deformations of the curve X ⊂ P 3 = Proj(k[x, y, u, w]) are described in Lemma A.5, and are given by the maximal minors of x + a 2 w a 7 y + a 6 w u + a 12 x + a 11 y + a 10 u + a 9 w y + a 1 w u + a 5 x + a 4 y + a 3 w x + a 8 w , over the polynomial ring k[a 1 , . . . , a 12 ]. The morphism i : X −→ P 3 is determined by the linear map i * (x, y, z, w) = (x, y, 0, w). This map is deformed by perturbing all possible entries. We need 16 variables b 1 , . . . , b 16 to perturb every component as a linear expression in x, y, u, w.
As we are allowed to perform coordinate transformations in the source of i, we can use the invertible map sending x to the first component of the deformed map, y to the second, u to u and w to the fourth component. By performing these transformations the map simplifies to the linear map i * R determined by i * R (x, y, z, w) = (x, y, b 9 x + b 10 y + b 11 w + b 12 u, w) . The matrix displayed above, describing the embedded deformations of the curve, then changes. But, after a linear change in the a i 's, depending on the b j 's we can take this matrix to be of the same form. The remaining available transformations can be used to make the (1, 3)-entry of the matrix displayed into u. After another linear change of coordinates in the a i 's the infinitesimal deformations are given by the matrix (2.13.1) and the morphism i R . These formulas define a deformation over the completion of the polynomial ring k[a 1 , . . . , a 8 , b 9 , . . . , b 12 ] in the ideal (a 1 , . . . , a 8 , b 9 , . . . b 12 ).
The next two lemmas will be used to describe the space of CM curves, with polynomial 3t + 1, in the plane and in P n with n > 3 respectively. Lemma 2.14. Let X ⊆ P 3 be the closed subscheme given by the graded ideal I = (u 2 , uy − x 2 , xu) ⊆ k[x, y, u, w], and let i 1 : X −→ P 2 be the morphism (2.12.1). The completion of the local ring of the point (X, i 1 ) in CM 3t+1 P 2 is given by the power series ring R 1 = k[[a 1 , . . . , a 8 ]]. The universal family over Spec(R 1 ) is given by the pair (X R 1 , i 1 ), where X R 1 is given by the ideal J ⊆ R 1 [x, y, u, w] generated by the maximal minors of (2.13.1) and the morphism i R 1 : X R 1 −→ P 2 R 1 , is the one obtained from i 1 . Proof. We proceed as in the proof of Lemma 2.13. Now we perturb the map i 1 in all possible ways, and we can use coordinate transformations in the source to undo these perturbations.
Lemma 2.15. Let X ⊆ P 3 be the closed subscheme given by the graded ideal I = (u 2 , uy − x 2 , xu) ⊆ k[x, y, u, w] as in Lemma 2.14, but now let i 2 : X −→ P n = Proj(k[x, y, w, z 1 , . . . , z n−2 ]) (n > 3) be the morphism determined by the linear map that sends z i to 0 (for i = 1, . . . , n − 2) and is the identity on x, y and w.
The completion of the local ring of the point (X, i 2 ) in CM 3t+1 P n is given by the power series ring The universal family over Spec(R 2 ) is given by the pair (X R 2 , j 2 ), where X R 2 is given by the maximal minors of (2.13.1) and the morphism i R 2 : X R 2 −→ P n is determined by the linear map that is the identity on x, y and w, and that sends (z 1 , . . . , z n−2 ) to Proof. Similar to the proof of Lemma 2.13.
Proposition 2.16. The space CM 3t+1 P 3 is smooth and irreducible of dimension 12. Proof. Let (X, i) be a k-point of CM = CM 3t+1 P 3 By Proposition 2.11 we may assume that X is embedded in some P 3 and that the morphism i is the restriction of a rational map h : P 3 P 3 . In particular, X itself is a curve on the twisted cubic component. Furthermore, the same holds for all infinitesimal deformations.
If i : X −→ P 3 is a closed immersion, then we may assume that h is the identity and remains so under deformation. Consequently the infinitesimal deformations of i : X ⊂ P 3 are the embedded deformations of X. Therefore the tangent space at the point (X, i) is isomorphic to the tangent space to the twisted cubic component in the point X, which has dimension 12 [PS85,p. 766].
Assume now that i : X −→ P 3 is not a closed immersion. By perturbing the coefficients defining the rational map h we can deform the morphism i : X −→ P 3 to a closed immersion. Thus any point on CM is the specialization of a twisted cubic in P 3 . It follows that the space CM is irreducible, and consequently to prove the proposition it suffices to show that the tangent space at any point has dimension 12.
As the image i(X) = D spans a plane, the map h is the projection from a point Q. Because i fails to be an isomorphism over exactly one point of D, the point Q does not lie on two different tangent or secant lines. Considered as a point on the twisted cubic component, the curve X ⊂ P 3 cannot be the most degenerate curve, which is a triple line given by the square of the ideal of a line [Har82, Section 1.b]. Such a curve has everywhere embedding dimension 3, so is not locally planar. Every other Cohen-Macaulay curve on the twisted cubic component degenerates to triple line on a quadric cone. For such a curve the center Q of the projection does not lie on the tangent plane to the cone containing the line. All maps i : X −→ P 3 with X a triple line on a cone and i generically an isomorphism are projectively equivalent, so it suffices to compute the tangent space in one specific example, for which we take the curve (X, i) of Lemma 2.13. By specialization the dimension of the tangent space at any point is bounded from above by the dimension of the tangent space at this (X, i), which is 12.
Corollary 2.17. For n > 3, the space CM 3t+1 P n is smooth and irreducible of dimension 4n. Proof. Let i : X −→ P n be a CM-curve. If i is a closed immersion it follows from Lemma 2.10 that the curve X is contained in some P 3 , and from Proposition 2.11 that infinitesimal deformations of i(X) ⊂ P n are the embedded deformations of X. The dimension of the tangent space is the dimension of H 0 (X, N X/P n ), which can be computed as the sum of the dimensions of H 0 (X, N X/P 3 ) and H 0 (X, N P 3 /P n ⊗ O X ), where N stands for the normal sheaf. The dimension is 12 + 4(n − 3) = 4n.
If i is not a closed immersion then we have by Proposition 2.9 that the image is planar. Arguing as in the proof of Proposition 2.16 we see that the space CM is irreducible and that to conclude the proof it suffices to compute the dimension of the tangent space in the explicit example of Lemma 2.15. There the dimension is 4n.
Remark 2.18. We note that the space CM 3t+1 P n is smooth and irreducible of dimension 4n, for all n ≥ 3. The case n = 2 will be described quite explicitly in Section 4. In particular, the space CM 3t+1 P 2 is smooth of dimension 8. This can also be shown using Lemma 2.14.
Remark 2.19. For CM-curves (X, i) with Hilbert polynomial 2t + 2 the analogue of Proposition 2.9 holds: if the map i is not a closed immersion, then the image D = i(X) is a singular plane conic, and there is exactly one point P ∈ D over which the morphism i fails to be an isomorphism. If i : X −→ P 3 is a closed immersion, then the curve is not arithmetically Cohen-Macaulay, but one can still show that CM 2t+2 P 3 is smooth and irreducible of dimension 8.

Plain double points
In this section we focus on CM-curves where the non-isomorphism locus is the simplest possible. The following definition is motivated by Proposition 2.9.
Definition 3.1. Let i : X −→ P n be a CM-curve, and let D ⊆ P n denote its schematic image.
Remark 3.2. The definition of a plain double point requires the double point locus of a map i : X −→ P n to be as simple as possible. The point P is always a singular point of the image, as a birational morphism onto a smooth curve is an isomorphism (apply this argument to a suitable affine neighbourhood of P ). But the singularity might be of higher type. For instance, let X be three lines in P 3 meeting in one point, not lying in a plane, and let i : X −→ P 2 be a general projection. Then the image D = i(X) consists of three lines in the plane through one point P . The point P ∈ D is a plain double point of the CM-curve i : X −→ P 2 . However the singularity P ∈ D is a triple point and not a (planar) double point.

The condition (R.C.)
In the following the Ring Condition (R.C.) of De Jong and Van Straten [dJvS90] plays an important role. We recall the setup. Let (R, m) be a local ring, let A be a local R-algebra, flat and Gorenstein over R.
is an isomorphism.
A submodule C of a ring A, that is an ideal, satisfies the Condition (R.C.) if the natural inclusion map (3.4.1) is an isomorphism.
Proposition 3.5. Let (X R , i R ) ∈ CM P 2 (Spec R) with R a local ring. Let i : X −→ P 2 denote the induced curve over the closed point in R, and assume that P ∈ P 2 is a plain double point. Let ξ ∈ P 2 R be the image of the point P under the natural inclusion P 2 ⊆ P 2 R . Then ξ is a point on the schematic image D R ⊆ P 2 R of X R , and we have a presentation where s and t are elements in the maximal ideal of O P 2 R ,ξ and g and f lie in the ideal n R = (s, t). Furthermore, we have the equality of ideals Proof. Let n denote the maximal ideal of the local ring A = O D,P , and set B = i * O X,P . It follows that n equals the annihilator ideal Ann A (B/A), which equals the conductor ideal {a ∈ A | aB ⊆ A}. The fact that B is a ring gives by Proposition 3.4 that B = Hom A (n, A) is isomorphic to Hom A (n, n).
We may assume that the plain double point P is (0 : 0 : 1), so the maximal ideal n is the ideal (x, y). As P is a plain double point the A-module B is generated by two elements, say 1 and u. The element u corresponds to an endomorphism µ u : n −→ n. We get that µ u (x) = −ḡ and µ u (y) = −f for somef andḡ in n. In other words we have that ux +ḡ = 0 and uy +f = 0. We get a surjective map A ⊕ A −→ B given by (a 1 , a 2 ) → a 1 + a 2 u. The kernel contains the elements (ḡ, x) and (f , y) and they in fact generate the kernel: the image of (a, 1) is a + u, which is not zero as u A. We have therefore the presentation As B is a Cohen-Macaulay O P 2 ,P -module of codimension one, generated by 1 and u, it has a free resolution of length one. Therefore we have a presentation over O P 2 ,P , obviously given by the same matrix where we use the same notation for elements in O P 2 ,P as for elements in A. The determinant xf − yḡ of the matrix defines the scheme-theoretic image i(X) = D locally around P . By assumption X R −→ Spec(R) is flat and it follows that the presentation over O P 2 ,P can be lifted to a presentation over O P 2 R ,ξ , where ξ is the image of P , see e.g. [Art76]. Thus, there are elements s, t, g, f in O P 2 R ,ξ that specialize to x, y,ḡ,f in O P 2 ,P , respectively, giving us the exact sequence , which is the principal ideal (sf − gt). As the map i R is generically an embedding, any other element in the kernel would have to be supported at ξ, but as the ideal (sf − gt) has no embedded components the equality of ideals I = (sf − tg) follows. So the matrix appearing in (3.5.3) also defines a presentation of i R * O X R ,ξ as O D R ,ξ -module and the ideal n R = (s, t) is the annihilator ideal n R = Ann D R (i R * O X R ,ξ /O D R ,ξ ). Furthermore, as the elements s and t specialize to x and y we have that O P 2 ,ξ /n R = R. Thus n R is flat. By the condition (R.C.) the element u ∈ i R * O X R ,ξ corresponds to an endomorphism µ u : n R −→ n R , with µ u (s) = −g and µ u (t) = −f . This means that g, f ∈ n R .
Proof. All three assertions are established in the proof of the above proposition.
Remark 3.7. The Ring Condition (R.C.) for the ideal n or n R is equivalent to the condition that the entries of the first row of the matrix in (3.5.2) or (3.5.1) lie in the ideal n, respectively n R . This is the content in this special case of Catanese's Rank Condition (R.C.) [Cat84]; for the terminology see also [dJvS90, Remark 1.13].

Singular sections of cubics
We give in this section an explicit description of the space of CM-curves in the plane having Hilbert polynomial 3t + 1. The space of such CM-curves is identified with plane cubics together with a singular section.

Critical locus
Let ϕ : X −→ S be a flat morphism of schemes, of pure relative dimension d. The critical locus of the morphism ϕ is the closed subscheme C(ϕ) ⊆ X given by the ideal sheaf Fitt d (Ω 1 X/S ), where Ω 1 X/S is the sheaf of differentials, see [Tei77]. A section σ : S −→ X of the morphism ϕ : X −→ S is a singular section if it factorizes through the critical locus C(ϕ). Thus, in the commutative diagram

Singular cubics
Let SC denote the functor parametrising cubics in P 2 with a singular section. That is, the S-valued points of SC are pairs (D, σ ) where D ⊆ P 2 S = P 2 × S is a flat family of cubics over S, and where σ : S −→ D is a singular section.
Proof. Let S be a scheme, and let s : S −→ C(ϕ) be a morphism. The morphism s is determined by a pair (σ , t), where σ : S −→ P 2 and t : S −→ Hilb 3t P 2 are morphisms that together factorize through the closed subscheme C(ϕ) ⊆ P 2 × Hilb 3t P 2 . The morphism t : S −→ Hilb 3t P 2 is equivalent with having a cubic D S ⊆ P 2 S = P 2 ×S, flat over S. The morphism σ : S −→ P 2 is the same as having a section of P 2 S −→ S. Now, as our pair (σ , t) is a point of the critical locus C(ϕ) it means that the partial derivatives of the cubic vanish over σ . In other words the section σ : S −→ P 2 S factors through the critical locus of the cubic D S −→ S. And conversely, given a flat family D ⊆ P 2 S of cubics over S with singular section σ : S −→ D ⊆ P 2 S we obtain morphisms σ : S −→ P 2 and t : S −→ Hilb 3t P 2 that together factorize through C(ϕ). From Example 4.2 we get that the ideal of the critical locus is locally defined by the partial derivatives of the cubic. It follows from local calculations that C(ϕ) is smooth of dimension 11 − 3 = 8.
Proposition 4.5. The functor CM 3t+1 P 2 of CM-curves in P 2 having Hilbert polynomial 3t + 1 is isomorphic to the functor SC of cubics with singular section. In particular CM 3t+1 P 2 is represented by the scheme C(ϕ), given as the critical locus of the universal family of cubics in the plane.
Proof. First we define a morphism φ : CM 3t+1 P 2 −→ SC. Given a scheme S and i S : X S −→ P 2 S an S-valued point of CM 3t+1 P 2 , we let D S ⊆ P 2 S be the schematic image of X S and N S ⊆ D S the subscheme defined by the annihilator of i S * O X S /O D S . Let R be the local ring of a closed point of S and denote by i : X −→ P 2 the curve over this closed point. By Proposition 2.9 the image D = i(X) is a cubic curve and there is one unique point P ∈ D where the induced map i : X −→ D is not an isomorphism. The point P is a plain double point. By Corollary 3.6 we have that the schematic image D R ⊆ P 2 R is flat over R, and that the subscheme N R ⊆ D R defined by the annihilator of i R * O X R /O D R , determines a singular section of D R −→ Spec(R). Thus (D S , N S ) is a flat family of cubics in P 2 with a singular section.
Next we define a morphism θ : SC −→ CM 3t+1 P 2 . Let (D S , N S ) be a flat family of cubics in P 2 with a singular section. We have that D S is given by a cubic form Q ∈ O S [x, y, w]. Consider an open affine set U ⊂ S on which the section N U is given by two linear independent forms s and t. As the section is singular we have that the cubic form Q ∈ O U [x, y, w] can be written as Q = s 2 f 1 + st(f 2 − g 1 ) − t 2 g 2 . Consider now a matrix factorisation of Q (4.5.1) M = g 1 s + g 2 t f 1 s + f 2 t s t .
We can view the matrix M as the presentation matrix of a sheaf F U on P 2 U . That is, we have the global presentation With generators 1, u of F U we have the relations (u + g 1 )s + g 2 t = 0 and f 1 s + (u + f 2 )t = 0. By formally eliminating s and t we obtain a third relation (u + f 2 )(u + g 1 ) = g 2 f 1 . These relations can be written as the maximal minors of the matrix The maximal minors define an arithmetically Cohen-Macaulay curve X U ⊂ P 3 U = U × Proj(k[x, y, u, w]), flat over U . For each point in U , the fiber is a curve with Hilbert polynomial 3t + 1, and the curve does not pass through the point (0 : 0 : 1 : 0). Projection from (0 : 0 : 1 : 0) induces a map i U : X U −→ P 2 U , that is an isomorphism onto its image D U outside the section N U .
A different choice of the sections s and t and of the matrix factorization gives a curve isomorphic to X U . The image of X U in P 2 U and the section N U are independent of these choices, and it follows that the curves defined for different open sets U ⊆ S glue together to a curve i S : X S −→ P 2 S , and thus an S-valued point of CM 3t+1 P 2 . We next show that the two morphisms constructed are inverse of each other. For this we may assume that the base is the spectrum of a local ring R. Let (D R , N R ) be a flat family of cubics with a singular section. Let i R : X R −→ P 2 R be the CM-curve we get by applying the morphism θ to the pair (D R , N R ). From the construction we have that φ • θ is the identity, and we verify that θ • φ is the identity. Outside the section N R the two curves X R and D R are isomorphic, so in particular X R \ i −1 R (N R ) is determined by the pair (D R , N R ). Over the closed fiber we have that the non-isomorphism locus is one point P ∈ P 2 , and we let ξ ∈ P 2 R be the image of P under the natural inclusion. By Proposition 3.5 we have that the ideal n R of the section N R ⊂ D R is contained in the ideal defining ξ. Let A = O R,ξ and B = (i R * O X R ) ξ . The last statement of Proposition 3.5 tells us that the ideal n R is the annihilator ideal Ann A (B/A). Hence n R is the dual module Hom A (B, A). By Proposition 3.4 we then have that the A-algebra B is the endomorphism ring Hom A (n R , n R ). Thus the structure sheaf i R * O X R is determined, up to isomorphism, by the pair (D R , N R ). It follows that θ • φ is the identity.
Remark 4.6. The proof could have been shortened using Zariski Main Theorem. We have chosen to give an explicit proof that we believe is more illuminating.
Remark 4.7. There is a natural map from C(ϕ) to Hilb 3t P 2 = P 9 , obtained by composing the closed immersion with the projection. This is a map from the space of cubics with a singular section SC to the Hilbert scheme forgetting the singular section. The image will be the set of singular cubics in the plane. Outside the locus of cubics with multiple components the map is the normalization morphism, see [Tei77, Theorem 5.5.1]. Over the set corresponding to multiple components the map is a proper modification.

Remarks on conics
It turns out that the situation with CM-curves in P 2 having Hilbert polynomial 2t + 2 can be treated quite analogously to the situation with twisted cubics. For readability we did not merge those similar arguments in the above text. We claim that they show that the space CM 2t+2 P 2 of CM-curves in the plane having Hilbert polynomial 2t + 2 is isomorphic to the space of plane conics with a singular section.

The space of twisted cubics
This section contains our main contribution that identifies the space of CM-curves in P 3 having Hilbert polynomial 3t + 1 with one specific component of the Hilbert scheme of twisted cubics in P 3 . We also treat the case n > 3.
Lemma 5.1. Let i R : X R −→ P 3 R be an element of CM 3t+1 P 3 (Spec(R)), where R is a local ring. Suppose that the induced map i : X −→ P 3 over the closed point in R is not a closed immersion. Then the closed subscheme Z ⊆ P 3 R defined by the 0 th Fitting ideal sheaf of i R * O X R is flat over Spec(R), with Hilbert polynomial 3t + 1.
Proof. The image of the CM-curve i : X −→ P 3 over the closed fiber is by Proposition 2.9 a plane cubic. We may assume that the image lies in the plane Π = {z = 0} ⊂ P 3 = Proj(k[x, y, z, w]) and that the plain double point is (0 : 0 : 0 : 1). From Proposition 2.11 it follows that we may assume that the curve X lies in P 3 = Proj(k[x, y, u, w]) and that the morphism i is induced by the linear map that sends (x, y, z, w) to (x, y, 0, w). In particular we may assume that the curve X does not pass through the point (0 : 0 : 1 : 0). We compose the map i R : X R −→ P 3 R with the rational projection P 3 R Π R . This provides us with an element of CM 3t+1 Π (R) and the image of X R is by Proposition 4.5 a cubic D R ⊂ Π R together with a singular section N R . After a change of coordinates we may assume that the section is given by s = x and t = y. Then the maximal minors of (4.5.2) (with f 2 = 0) that generate the ideal of X R in P 3 = Proj(k[x, y, u, w]) are where g = g 1 x + g 2 y and f = xf 1 . The cubic D R ⊂ Π R is given by R is then obtained from the linear map i * that sends z to αw + βu + γy + δx with α, β, γ and δ in the maximal ideal of R. By a coordinate change in the image, making z − αw − γy − δx the new z-coordinate we bring the map into the final form: i * sends z to βu and is the identity on x, y and w.
We now compute a presentation of X R over P 3 R . In (4.5.1) we have a presentation of X R over P 2 R . We identify the plane with the subscheme in P 3 R defined by z = 0. We have that i * O X R is generated by two elements 1 and u over O P 3 R . To obtain a presentation we only need to add the relations coming from the action of z. We have that z · 1 = βu, and therefore z · u = βu 2 . We rewrite βu 2 using the relation u 2 + ug 1 − f 1 g 2 = 0. Therefore we have the presentation A computation shows that the 0 th Fitting ideal I = Fitt 0 (i * O X R ) is generated by four elements Q, F 1 , F 2 , and F 3 , where Q = Q(x, y, w) is our recurring cubic, and where As R[x, y, w]/(Q) is a flat family of cubics the R-module of degree n forms is locally free of rank 3n. From the leading terms of F 1 , F 2 and F 3 it follows that the R-module of degree n forms in the graded quotient ring R[x, y, z, w]/ (Q, F 1 , F 2 , F 3 ) is locally free of rank 3n + 1, adding only the free component with basis zw n−1 to the forms determined by the cubic Q alone. Therefore Z is flat over R.

The twisted cubic component
The Hilbert scheme Hilb 3t+1 P 3 of closed subschemes in P 3 having Hilbert polynomial 3t + 1 consists of two smooth components. This was proven by Piene and Schlessinger [PS85] when the base field has characteristic different from 2 and 3, but their result is valid in any characteristic, see Proposition A.1. One of these components is 12 dimensional and contains an open subset that parametrizes twisted cubics in P 3 . We refer to the component H as the twisted cubic component.
To show that the morphism φ is well-defined it suffices to check it over an affine base scheme S = Spec(R). If the map i R : X R −→ P 3 R is a closed immersion there is nothing to prove as the 0 th Fitting ideal is the the defining ideal of the image of the closed immersion. If the map is not a closed immersion it follows from Lemma 5.1 that the 0 th Fitting ideal defines a flat subscheme with appropriate Hilbert polynomial. We have therefore a morphism φ from CM 3t+1 P 3 to the Hilbert scheme Hilb 3t+1 P 3 . As CM 3t+1 is irreducible it follows that the morphism factorizes through the twisted cubic component H and gives the sought morphism. Let Z ⊆ P 3 be a closed subscheme corresponding to a point on the twisted cubic component H. There is a divisor δ on H corresponding to singular planar cubics together with a spatial embedded point. If Z corresponds to a point on H but not on the divisor δ then the closed immersion i : Z −→ P 3 is a CM-curve. If Z is a point on the divisor, then it is a plane cubic, lying in a plane Π ⊂ P 3 , with an embedded point at a singular point P of the cubic, see ([PS85, Lemma 2]). By Proposition 4.5 there is a (unique) CM-curve i Π : X −→ Π such that the image of X is the singular cubic and the map i Π is not an isomorphism onto its image at P . Let i be the composed map i : X −→ Π ⊂ P 3 . Then the image by φ of the curve (X, i) is Z ⊂ P 3 . More explicitly, the scheme Z is projectively equivalent to the scheme given by the ideal I = (xz, yz, z 2 , Q) where Q = Q(x, y, w) is a cubic form singular at (0 : 0 : 1) (see [PS85, Lemma 2]). As in the proof of Lemma 5.1 the curve X is given by the maximal minors of the matrix (4.5.2) and we have a presentation (5.1.1) with β = 0. The 0 th Fitting ideal is the ideal (Q, z 2 , zx, zy). It follows that the morphism φ : CM −→ H is bijective. Because the spaces are isomorphic on an open dense subset, the isomorphism follows from Zariski's Main Theorem [Mum88, Chapter III, §9] as both spaces are smooth of dimension 12.
Remark 5.4. Freiermuth and Trautmann studied the moduli scheme of stable sheaves supported on cubic space curves [FT04]. In characteristic zero there exists a projective coarse moduli space M p X for semi-stable sheaves on a smooth projective variety X with a fixed Hilbert polynomial p. For p(t) = 3t + 1 and X = P 3 all sheaves in M = M 3t+1 P 3 are stable. The result of [FT04] is that the projective variety M consists of two nonsingular, irreducible, rational components M 0 and M 1 of dimension 12 and 13, intersecting transversally in a smooth variety of dimension 11, see also Appendix B.
The component M 0 is isomorphic to the twisted cubic component H of the Hilbert scheme. The identification also uses Fitting ideals. If i : X −→ P 3 is a CM-curve, with Hilbert polynomial 3t + 1, then the module i * O X is a stable sheaf supported on a cubic. Thus, by forgetting the algebra structure of i * O X we get that our morphism φ : CM 3t+1 P 3 −→ H factorizes through the moduli scheme of stable sheaves. Example 5.5. The following example shows that the 0 th Fitting ideal does not always give a morphism from CM to the Hilbert scheme. Indeed, we show that here the family of closed subschemes determined by the 0 th Fitting ideal is not flat.
We start with a genus 2 curve embedded with the linear system |5P |, where P is a Weierstrass point. More precisely, we look at the curve X ⊂ P 3 given by the homogeneous ideal generated by the maximal minors of x u y 2 + w 2 y x u 2 .
The projection from X to the plane P 2 in the coordinates x, y, w is finite. Let another P 3 be given by the homogenous coordinate ring k[x, y, z, w], and consider the family of maps i t : X t X −→ P 3 , t ∈ T = A 1 , determined by the linear map sending (x, y, z, w) to (x, y, tu, w).
In the affine chart {w = 1} we get that i t * O X has presentation The 0 th Fitting ideal is then the ideal generated by z 3 − t 3 x(y 2 + 1), z 2 x − t 2 y(y 2 + 1), zx 3 − ty 2 (y 2 + 1), x 5 − y 3 (y 2 + 1), The family determined by this ideal is not flat, as yz − tx 2 is a t-torsion element. For t 0 we have the generator yz − tx 2 in the ideal, making the three first generators on the last row above superfluous. The family above gives an T = A 1 -valued point (X T = X × A 1 , i T ) of CM 5t−1 P 3 . Then by taking the 0 th Fitting ideal of i T * O X T we get a closed subscheme Z ⊆ P 3 A 1 which is not a flat family, and in particular the Hilbert polynomial of a fiber is not constant.

Higher codimension
The twisted cubic component H n of the Hilbert scheme Hilb 3t+1 P n for n > 3 has been described by Chung and Kiem [CK11]; their proof works in any characteristic. The component H n is isomorphic to a component of the relative Hilbert scheme of the P 3 -bundle PU −→ G n+1 4 , where U is the universal rank 4 vector bundle on the Grassmannian G n+1 We remark that the construction with the Fitting ideal does not give a morphism from CM 3t+1 P n to Hilb 3t+1 P n if n > 3. Indeed, if the image of a CM-curve X −→ P n is a planar curve then the 0 th Fitting ideal gives a scheme with Hilbert polynomial 3t + n − 2. By a coordinate transformation we may assume that the plane containing the image i(X) is given by z 1 = · · · = z n−2 = 0 and that the non-isomorphism locus on the image i(X) is the point (0 : · · · : 0 : 1). A presentation of i * O X as O P n -module is then given by the matrix The 0 th Fitting ideal is the ideal I = (z 2 i , z i z j , z i x, z i y, yg − xf ) for 1 ≤ i ≤ n − 2 and i < j ≤ n − 2. There is a P n−3 of P 3 's containing the plane z 1 = · · · = z n−2 = 0 and each of these P 3 's contains a subscheme Z with Hilbert polynomial 3t + 1 with the planar singular cubic as 1-dimensional subscheme. To study this morphism σ we first describe CM in a neighbourhood of a curve i : X −→ P n whose image is a planar curve. We may assume that the plane is given as z 1 = · · · = z n−2 = 0, that X is given by an ideal I in k [x, y, u, w], and that the map i is given by i * (x, y, z 1 , . . . , z n−2 , w) = (x, y, 0, . . . , 0, w). It follows from Lemma 2.15 and by arguing as in the proof of Lemma 5.1 that we may assume that a neighborhood of the curve i : X −→ P n is given by perturbing the ideal I ⊂ k [x, y, u, w] and perturbing the map i * to the map j * that sends (x, y, z 1 , . . . , z n−2 , w) to (x, y, b 1,1 x+b 1,2 y+b 1,3 w+b 1,4 u, . . . , b n−2,1 x+b n−2,2 y+b n−2,3 w+b n−2,4 u, w). The locus of CM-curves with planar scheme-theoretic image is locally given by the equations b 1,4 = b 2,4 = · · · = b n−2,4 = 0.
An affine chart of the Grassmannian is obtained by choosing four global sections of H 0 (P n , O P n (1)) that form a basis of H 0 (X, i * O P n (1)). The affine chart is then the affine space representing linear maps from the remaining n − 3 global sections to H 0 (X, i * O P n (1)). By choosing the global sections x, y, w, z 1 the universal family over the affine chart is then the map that sends z i → λ i,1 x + λ i,2 y + λ i,3 w + λ i,4 z 1 , for every i = 2, . . . , n − 2. The condition that the map j * factors through the quotient map on the affine chart of the Grassmannian is that b k,m = λ k,m + λ k,4 b 1,m for 2 ≤ k ≤ n − 2, m = 1, 2, 3 and b k,4 = λ k,4 b 1,4 for 2 ≤ k ≤ n − 2. The last n − 3 equations show that the morphism σ is the blow up of the locus of CM-curves with planar scheme-theoretic image.
Let U be the universal quotient bundle over G. By [CK11] the Hilbert scheme Hilb 3t+1 P n is isomorphic to the relative Hilbert scheme Hilb 3t+1 PU −→ G, with fibres isomorphic to Hilb 3t+1 P 3 . We define a morphism from CM to the Hilbert scheme

The Hilbert scheme component with two skew lines
The Hilbert scheme Hilb 2t+2 P 3 of closed subschemes in P 3 over a field of characteristic zero, having Hilbert polynomial 2t + 2 consists of two smooth components. The smoothness of these components was observed in [Har82], and a proof was given by Chen, Coskun, and Nollet in [CCN11]. One of these components H 3 is smooth of dimension 8 and a general point correspond to a pair of skew lines. We claim that with the arguments presented in the present article, one can prove that we have a morphism φ : CM 2t+2 P 3 −→ H 3 sending a S-valued point (X, i) to the closed subscheme in P 3 × S defined by the 0 th Fitting ideal sheaf Fitt 0 (i * O X ). The morphism is an isomorphism.

A. The Hilbert scheme of twisted cubics
The main purpose of this section is to prove the following result. Remark A.2. The statement about the structure of the Hilbert scheme for algebraically closed fields of characteristic different from 2 and 3 is found [PS85]. The reason for avoiding these characteristics lies in the deformation computation in Section 5 of their paper; this is not needed for the results in the rest of the paper. Our characteristic free Lemma A.8 given below, replaces Lemma 6 in loc. cit., from where the proposition then follows. A detailed explanation of this type of deformation computation is given in [Ste95], which furthermore contains an easier example of two intersecting lines with an embedded point at the origin, relevant for the (2t + 2)-case.

A.4. Tangent space calculations
We start by reproving the following, known, explicit description of the following open affine chart of the Hilbert scheme Hilb 3t+1 of closed subschemes in P 3 with Hilbert polynomial 3t + 1, a result we relied on in Section 2. x + a 2 w a 7 y + a 6 w u + a 12 x + a 11 y + a 10 u + a 9 w y + a 1 w u + a 5 x + a 4 y + a 3 w x + a 8 w generate an ideal I(a) ⊂ A[x, y, u, w] that determines the restriction of the universal family to Spec(A).
Proof. The first order deformations are determined by the global sections of the normal sheaf H 0 (Z, N Z/P 3 ).
It is convenient to work in the affine chart {w = 1}. As the ideal is determinantal, we can compute as described in [Sch73]. The sections of the normal sheaf are then generated by the following six deformations induced by perturbing the matrix x + ε 11 ε 12 u + ε 13 y + ε 21 u + ε 22 x + ε 23 .
We describe an infinitesimal deformation by its action on the vector (u 2 , yu − x 2 , xu) of generators of the ideal. Written out this action reads We can also multiply the generators by linear functions. We have to consider the action on the generators modulo the ideal (u 2 , yu − x 2 , xu). We find that u ∂ . We therefore get 6 additional first order deformations. As deformations of determinantal varieties are unobstructed we homogenize with respect to the variable w and write a 12-dimensional family by choosing appropriate representatives. With new names for the deformation variables this can be presented as the claimed maximal minors.
Proposition A.6. Let Z ⊆ P 3 be the closed subscheme given by the graded ideal I = (z 2 , zx, zy, Q) ⊆ k[x, y, z, w], where Q = Q(x, y, w) is a cubic form. If Q is singular at (0 : 0 : 1) then the tangent space to the Hilbert scheme at Z has dimension 16. If Q is smooth at (0 : 0 : 1), then the tangent space has dimension 15.
Proof. Write Q = Q(x, y, w) = xf − yg in A = k[x, y, z, w]. We then have the free resolution where ϕ = (zx, zy, z 2 , Q) is the map given by the generators of the ideal. The relation matrix R is then Again it is convenient to work in the affine chart {w = 1}. We compute deformations as described in [Art76, Section I.6] and [Ste95]. We obtain generators for the global sections of the normal sheaf N Z/A 3 as the syzygies of the transpose of the relation matrix R, but computed modulo the ideal I = (zx, zy, z 2 , Q).
We give the generators of the normal sheaf by their action on the vector ϕ of generators of the ideal. If Q = xf − yg is singular at (0 : 0 : 1) then neither f nor g have non-zero constant term. It follows that generators of the normal sheaf are As the degree of a perturbation can be at most that of the element of ϕ, so 2, 2, 2, 3 respectively, and one has to compute modulo the ideal I, we get 8 additional deformations by multiplying the last four generators above with the variables x and y. It follows that the dimension of the Zariski tangent space to Hilb 3t+1 is 16. If Q has linear terms, the last generator (g, f , 0, 0) is not present, and (z, 0, 0, 0) is to be replaced by (z, 0, 0, f ), and (0, z, 0, 0) by (0, z, 0, −g). Then the dimension of the Zariski tangent space is 15.
Remark A.7. The ideal (zx, zy, z 2 , Q) with Q smooth at the origin gives a plane cubic through the origin with an embedded point at the origin, so on the curve. It is also possible to have a plane cubic and a point in its plane but not on the cubic. Such a curve is not a small deformation of a curve in the intersection of the two components of the Hilbert scheme. Consider the plane cubic z = x 3 + y 3 + w 3 = 0 and a point at (0 : 0 : t : 1). For t 0 the homogeneous ideal is generated by zx, zy, z(z − tw), zw 2 − t(x 3 + y 3 + w 3 ) but if we specialize to t = 0 we also need the equations x(x 3 + y 3 + w 3 ) and y(x 3 + y 3 + w 3 ). The result is the non-saturated ideal zx, zy, z 2 , zw 2 , x(x 3 + y 3 + w 3 ), y(x 3 + y 3 + w 3 ) .
the direction of the twisted cubic component and the variables c 13 , . . . , c 16 represent deformations in the direction of the other component.

B. The moduli space of stable sheaves
In this section we relate the moduli space of stable sheaves on P 3 with Hilbert polynomial 3t + 1 to the Hilbert scheme. The definition of (semi-)stability is recalled in [FT04, Section 2]. Note that they work over a fixed, algebraically closed field of characteristic zero. Every stable sheaf F on P 3 with Hilbert polynomial 3t + 1 has a free resolution of the form [FT04] 0 Any flat deformation of F is obtained by perturbing the matrices A and B to A and B such that A B = 0. We consider again the most singular point on the intersection of both components of M 3t+1 P 3 , which is the sheaf F = i * O X with (X, i) the CM-curve of Lemma 2.13. It has a presentation with matrices where againx = x + a 2 w,ỹ = y + a 1 w,z = z + a 11 w + a 9 x + a 10 y,ã 3 = a 3 w + a 4 y + a 5 andã 6 = a 6 w + a 7 y.
The map M 3t+1 P 3 −→ Hilb 3t+1 P 3 is given by the identity map on the variables a 1 , . . . , a 11 , b 12 , c 13 , c 14 . Proof. Infinitesimal deformations of the matrices A and B can be found by direct computation, see also [FT04,Section 7]. Then one has to try to lift the equation AB = 0. The result is as stated. One has Remark B.2. The matrix A, restricted to the component c 13 = c 14 = 0, gives a presentation of i R * O X R with (X R , i R ) the universal family of Lemma 2.13. To see this we use a coordinate transformation from the coordinates in the Lemma. We setx = x + a 2 w,ỹ = y + a 1 w,z = z − b 11 w − b 10 y − b 9 x,ã 3 = a 3 w + a 4 y + a 5 x andã 6 = a 6 w + a 7 y. We furthermore putã 8 = a 8 − a 2 . Then the curve becomes in the chart {w = 1} xã 6 ũ y u +ã 3x +ã 8 and the morphism i * R is (x,ỹ,z) → (x,ỹ, b 12 u).