title: Unirational moduli spaces of some elliptic K3 surfaces
authors: Mauro Fortuna Michael Hoff Giacomo Mezzedimi
text: 
   We show that the moduli space of U⊕⟨ -2k ⟩-polarized K3 surfaces is unirational for k ≤ 50 and k ∉{11,35,42,48}, and for other several values of k up to k=97. Our proof is based on a systematic study of the projective models of elliptic K3 surfaces in ^n for 3≤ n ≤ 5 containing either the union of two smooth rational curves or the union of a smooth rational curve and an elliptic curve intersecting at one point.
  
  
  Kernel-based Active Subspaces with application to CFD problems
    using Discontinuous Galerkin method
      [
      
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  § INTRODUCTION
  
  
  By classical results and works of Mukai <cit.>, it is known that the moduli spaces of complex K3 surfaces of genus g ≤ 12 and g=13, 16, 18, 20 are unirational. This was later improved by Farkas and Verra <cit.> extending the unirationality result to K3 surfaces of genus g=14, 22 by using the connection to special cubic fourfolds. Recently, the moduli spaces of n-pointed K3 surfaces of genus g≤ 22 were studied systematically in <cit.>. It is then natural to ask the more general question about the unirationality of moduli spaces of lattice polarized K3 surfaces. Farkas and Verra <cit.> worked out the case of polarized Nikulin surfaces. The case of 2-elementary K3 surfaces was studied in <cit.>, and further results in this direction were obtained in <cit.> by using orbits of representation of algebraic groups. In the present article, we will restrict to the case of elliptic K3 surfaces of Picard rank at least 3.
  
  We recall that a K3 surface S is called elliptic if it admits a fibration S→^1 in curves of genus one together with a section. The geometry of elliptic surfaces can be studied via their realization as Weierstrass fibrations. By using this description in <cit.>, Miranda constructed the moduli space of elliptic K3 surfaces and showed its unirationality (it is actually rational by <cit.>). The Néron-Severi group of the very general elliptic K3 surface is isomorphic to the hyperbolic plane U, and it is generated by the classes of the fiber and the zero section of the elliptic fibration. 
  
  For an elliptic K3 surface of Picard rank 3, its Picard lattice is isomorphic to U⊕⟨ -2k ⟩ for some integer k≥ 1. We are interested in studying the moduli spaces ℳ_2k of U⊕⟨ -2k ⟩-polarized K3 surfaces. These moduli spaces are divisors in the moduli space of elliptic K3 surfaces. 
  
  The study of ℳ_2k was initiated in <cit.>, where the authors showed the unirationality of ℳ_2k for some values of k≤ 64. Moreover, in the same article it is proven that the Kodaira dimension of _2k is non-negative for k ≥ 176 (and for other smaller values until k=140). We extend the unirationality result using the computer algebra system Macaulay2. 
  We construct projective models of U⊕⟨ -2k ⟩-polarized K3 surfaces in ^n for 3 ≤ n ≤ 5 for several values of k, leading to the following theorem.
  
   
  The moduli space ℳ_2k is unirational for the following values of k: 
  
      {   6, 7, 9,   10, 12,…, 34, 36, …, 41, 43, …, 47, 49,   50,
                52,    53, 54,  59, 60, 61, 62,  64,  68,   69,   73,   79,   81,   94,   97}.
  
  
  
  Combining this with the previous work in <cit.>, that constructed the moduli spaces _2k using double covers and Weierstrass fibrations, we obtain a more complete result:
  
   
  The moduli space ℳ_2k is unirational for k≤ 50, k ∉{11,35,42,48} and for the following values of k: 
  
      {52,    53, 54,  59, 60, 61, 62,  64,  68,   69,   73,   79,   81,   94,   97}.
  
  
  
  
  Our strategy involves a systematic study of the projective models of U⊕⟨ -2k ⟩-polarized K3 surfaces in ^3, ^4 and ^5 containing either two smooth rational curves or an elliptic curve and a smooth rational curve meeting at one point. We observe that similar ideas could be used to find new projective models, allowing curves of higher genus or working in higher dimensional projective spaces. The first approach seems however to produce no new cases. On the other hand, our method leads to the unirationality of the moduli spaces of many other lattice polarized (non-elliptic) K3 surfaces of Picard rank at least 3; we have not included these ones in this article.
  
  This research leads to several follow-up questions. First of all, one would like to understand the Kodaira dimension of _2k for the values of k in the remaining gaps. Moreover, it would be of great interest to find connections between the spaces _2k and other known geometric objects. One such connection was found in <cit.>, where the authors showed that the moduli space _56 is birational to a ^1-bundle over the universal Brill–Noether variety 𝒲_9,6^1 parametrizing curves of genus 9 together with a pencil of degree 6. Their strategy was to use the relative canonical resolution of these curves on rational normal quartic scrolls. We hope that our methods can be used to reveal new reincarnations of the moduli spaces _2k since we provide explicit methods to study such elliptic K3 surfaces and the geometry of their moduli spaces.
  
  The article is organized as follows. In Section <ref> we explain the general strategy to prove the main theorem. More precisely, we first find an exhaustive list of possible projective models for U ⊕⟨ -2k ⟩-polarized K3 surfaces as complete intersection in ^3, ^4 and ^5. This is then used to construct a dominant rational map I ℳ_2k (as in diagram <ref>), where I is a unirational parameter space (cf. Section <ref>). Section <ref> contains the experimental data obtained with our program in Macaulay2, with some examples. Finally, we extend  these contructions in Section <ref> to the case of nodal elliptic K3 surfaces.
  
  All our constructions are implemented in Macaulay2 (see <cit.>) and are available at the authors' homepage <cit.>.
  
  
  
   §.§ Acknowledgements
   We would like to thank Klaus Hulek and Matthias Schütt for useful discussions and for reading an early draft of this manuscript. We also thank the anonymous referee for carefully reading the paper and suggesting several improvements. The first author acknowledges partial support from the DFG Grant Hu 337/7-1.
  
  
  
  § GENERAL STRATEGY
   
  
  Our strategy is to construct projective models of K3 surfaces in ^n for 3 ≤ n ≤ 5 containing suitable pairs of curves. Let U be the hyperbolic plane. We focus on elliptic K3 surfaces of Picard rank at least 3, that is, U⊕⟨ -2k ⟩-polarized K3 surfaces for some k ≥ 1.
  We first recall the construction of the moduli space of lattice polarized K3 surfaces. Our main reference is <cit.>.
  
  For a hyperbolic lattice L embedding primitively in the K3 lattice Λ_K3=U^3⊕ E_8^2, the moduli space ℳ_L of L-polarized K3 surface is constructed as follows. Denote by T=L^⊥_Λ_K3 the orthogonal complement of L in Λ_K3.
  Let Ω_T be one of the two connected components of {w∈(T⊗) : (w,w)=0, (w,w)>0}, and consider the group O^+(T) of isometries of T preserving Ω_T. We denote by O^+(T) the subgroup of O^+(T) of isometries acting trivially on the discriminant group A_T=T^∨/T of T, where T^∨ is the dual lattice of T (see <cit.> for the precise lattice-theoretical definitions). Then ℳ_L is defined as the quotient O^+(T)\Ω_T. By the classical result of Baily and Borel <cit.>, ℳ_L is a quasi-projective variety of dimension (T)-2=20-(L).
  
  An L-polarized K3 surface is a pair (X,j), where X is a K3 surface and j:L↪NS(X) is a primitive embedding of L in the Néron-Severi group of X. The pair (X,j) defines the point in ℳ_L corresponding to the unique (up to scalars) nondegenerate holomorphic 2-form ω_X∈ j(L)^⊥_Λ_K3 on X. By the Torelli theorem, ℳ_L is the coarse moduli space of L-polarized K3 surfaces (cf. <cit.>).
  
  In our case we have L_2k=U⊕⟨ -2k⟩, so the moduli spaces ℳ_2k:=ℳ_L_2k we are interested in have dimension 17.
  
  For any fixed k, a very general U⊕⟨ -2k ⟩-polarized K3 surface X has NS(X)=U⊕⟨ -2k ⟩. In order to prove unirationality of some of these moduli spaces, we ask NS(X) to admit a basis given by a very ample polarization and two other smooth curves, 
  where we distinct  two cases: 
  
      
    Case 1: Two smooth rational curves meeting transversely at several points in general position;
      
    Case 2: A smooth rational curve and an elliptic curve meeting transversely at one point.
  
  
  The reason for this assumption is that we want to realize these K3 surfaces as complete intersections in ^n, 3≤ n≤ 5, and therefore we need a very ample divisor on X of square 2n-2 providing such a projective model. Since our K3 surfaces have Picard rank 3, we need two extra curve classes, and it turns out that allowing for the two cases listed above gives already a lot of flexibility, enabling us to show the unirationality of many of the moduli spaces ℳ_2k. Experimental data suggests that allowing two extra classes of curves of higher genus does not help in proving the unirationality of the remaining spaces ℳ_2k, so one probably needs to consider K3 surfaces with polarizations of higher degree in order to extend the results of this paper.
  
  Notice that in Case 2 the resulting K3 surfaces are automatically elliptic, since the elliptic curve induces the desired elliptic fibration, and the smooth rational curve becomes a section of the fibration.
  On the other hand, the K3 surfaces resulting from Case 1 are elliptic only in some cases, depending on the arithmetic of the Néron-Severi lattice resulting from the construction. 
  We explain in Section <ref> how to check whether the constructed K3 surfaces are actually elliptic.
  
  
  
   §.§ The construction
  
  
  We describe the construction in detail for Case 2. Let n,d,γ be integers such that 3≤ n ≤ 5, d ≥ 3 and  γ≥ 1. 
  
   
    Step 1: We construct a smooth elliptic curve E of degree d in ^n with a distinguished point p.
   
    Step 2: We construct a smooth rational curve Γ of degree γ intersecting E transversely only at the point p.
   
    Step 3: We choose (if it exists) a smooth K3 surface X in ^n of degree 2n-2 containing E∪Γ. 
  
  Then, we get a K3 surface X containing an elliptic curve E and a smooth rational curve Γ with the following lattice embedding
  
      [ 2n-2    γ    d;    γ   -2    1;    d    1    0 ]≅ U ⊕⟨ 2n-2 - 2dγ - 2d^2 ⟩↪NS(X).
  
  In particular, we set k = d^2 +dγ -n+1. We check case by case that X is a U⊕⟨ -2k ⟩-polarized K3 surface by showing that such a lattice embedding is always primitive (see Section <ref>). By abuse of notation, we denote by E and Γ the classes in NS(X) of the corresponding curves under this lattice embedding. Let _2k be the moduli space of U⊕⟨ -2k ⟩-polarized K3 surfaces. _2k is an irreducible variety of dimension 17.
  
  We can easily adapt the above strategy to Case 1. First, we construct a smooth rational curve Γ_1 ⊆^n of degree γ_1, together with m points p_1,…,p_m∈Γ_1. Then, we construct a second smooth rational curve Γ_2 intersecting Γ_1 transversely, precisely at p_1,…,p_m. Finally, Step 3 remains unchanged: we just choose (if it exists) a smooth K3 surface X in ^n containing Γ_1 ∪Γ_2.
  
  
  
  
  We compute in Macaulay2 that the constructed curves are smooth points of a component of the right dimension in the corresponding Hilbert schemes. By standard semicontinuity arguments (see e.g. <cit.>), we will perform our computations over a finite field (the main reason for doing this is that the computation is much faster over a finite field, but our constructions also work over the rationals). Finally, a dimension count shows that the construction dominates the corresponding moduli space _2k. We will present more details in the rest of the section.
  
  
  
   §.§ Unirationality
   
  
  The constructions described in Step 1, 2 and 3 can be realized as incidence varieties which are shown to be unirational.
  
  
  We denote by H_d,g,n the open subscheme of the Hilbert scheme parametrizing smooth irreducible curves of degree d and genus g in ^n. We notice that H_d,g,n is irreducible if g∈{0,1} and n∈{3,4,5} by <cit.>. Moreover, we can easily compute the dimension of H_d,g,n for g∈{0,1} by using the fact that every smooth curve C⊆^n of genus g ≤ 1 and degree d>0 is non-special, that is, H^1(𝒪_C(1))=0. Indeed, by using the Euler sequence and the defining sequence for the normal bundle of C⊆^n, we deduce that H^1(𝒩_C/^n)=0, and thus
  
      H_d,g,n=h^0(𝒩_C/^n)=
      (n+1)d+(n-3)    if  g=0
      
      (n+1)d    if  g=1.
  
  
  
  
  
   §.§ Step 1
  
  
  We include the proofs of the following classical results for the sake of completeness. 
  
  
   Let n,d≥ 3 be integers.
   The incidence variety
   
      H_d,1,n^1 := {(E,p) | p∈ E}⊆ H_d,1,n×^n
  
   of elliptic curves of degree d in ^n with a marked point is unirational. Its dimension is 
   
      H_d,1,n^1 =  H_d,1,n + 1.
  
  
  
  The moduli space _1,2 of elliptic curves marked with 2 points is rational by <cit.>. In order to construct an elliptic curve E of degree d≥ 3 in ^n together with a marked point p, we start with a plane cubic curve E' with two distinguished points p and q. The choice of a basis of the vector space V = H^0(E',Ø_E'(d q)) of dimension d yields a birational map
  
      E' ⟶ E⊂^d-1
  
  where the image E is a smooth elliptic curve of degree d (recall that all line bundles on E' of degree d are of the form Ø_E'(d q), see e.g. <cit.>). The choice of the basis is unirational since it is parametrized by an open subset of V^d. If d-1>n, then we project the curve E birationally to a smooth elliptic curve of degree d in ^n. If d-1<n, then we embed the ambient space ^d-1 into ^n in order to get again an elliptic curve of degree d in ^n. In both cases, we have to choose an appropriate linear subspace of ^n or ^d-1, and this choice is clearly unirational.
  
  Now consider the forgetful morphism ϕ :H_d,1,n^1→ H_d,1,n. ϕ is dominant, since every elliptic curve of degree d in ^d-1 arises as the image of a (plane or abstract) elliptic curve via a complete linear system of degree d, and every elliptic curve of degree d in ^n lies in a linear subspace of dimension d-1 if n>d-1.
  Since the preimage of a general point [E]∈ H_d,1,n is isomorphic to E itself, we conclude that H_d,1,n^1 =  H_d,1,n + 1.
  
  
  
  Let p_1,…,p_m ∈^n be a set of points spanning a linear subspace ^l ⊆^n. We say that p_1,…,p_m are in general position if they are in general position inside ^l. In particular, we have m ≤ l+1 ≤ n+1.  
  
  
   
  Let n≥ 3, γ≥ 1 and m ≤min{n+1,γ+1} be integers.
  Fix points p_1,…,p_m ∈^n in general position. The variety
   
      H_γ,0,n(p_1,…,p_m) := {Γ∋ p_1,…,p_m}⊆ H_γ,0,n
  
   of smooth rational curves of degree γ in ^n passing through p_1,…,p_m is irreducible and unirational. Moreover, it is non-empty, of dimension 
   
      (H_γ,0,n(p_1,…,p_m)) = (H_γ,0,n)-m(n-1).
  
  
  
  Since the variety of embeddings Emb_γ(^1,^n) of degree γ is rational and the morphism Emb_γ(^1,^n)→ H_γ,0,n sending a morphism to its image is dominant, we have that H_γ,0,n is irreducible and unirational. If p_1,…,p_m are points in ^n in general position, we consider the subvariety T of Emb_γ(^1,^n)× (^1)^m consisting of embeddings f:^1↪^n and m points x_1,…,x_m ∈^1 such that f(x_i)=p_i for all 1 ≤ i ≤ m. Since the conditions f(x_i)=p_i are linear in the coefficients of the polynomials defining the embedding f, and T dominates H_γ,0,n(p_1,…,p_m), we deduce that H_γ,0,n(p_1,…,p_m) is irreducible and unirational.
  
  In order to show that H_γ,0,n(p_1,…,p_m) is non-empty, fix a linear map h:^γ→^n and points q_1,…,q_m∈^γ in general position such that h(q_i)=p_i for all 1 ≤ i ≤ m. Since there is always a smooth rational normal curve C⊆^γ of degree γ passing through m ≤γ+1 points in general position, then Γ := h(C)⊆^n is the desired smooth rational curve.
  
  Finally, we have to compute (H_γ,0,n(p_1,…,p_m)). Let Γ∈ H_γ,0,n(p_1,…,p_m) be a smooth rational curve as constructed above, and consider the divisor D:=p_1+…+p_m over Γ. Let H be the restriction of the hyperplane class on ^n to Γ. Then the exact sequences
  
      0 →𝒯_Γ(-D) →𝒯_^n|_Γ(-D) →𝒩_Γ/^n(-D)→ 0
  
  and
  
      0 →𝒪_Γ(-D) →𝒪_Γ(H-D)^n+1→𝒯_^n|_Γ(-D) → 0,
  
  combined with the fact that H^1(𝒪_Γ(H-D))=0 (by Serre duality, as deg(𝒪_Γ(H-D))>-2), imply that H^1(𝒩_Γ/^n(-D))=0. Thus, the dimension of H_γ,0,n(p_1,…,p_m) coincides with χ(𝒩_Γ/^n(-D)), and a straightforward computation using the two previous exact sequences yields
  
      χ(𝒩_Γ/^n(-D))=(γ+1)(n+1)-4-m(n-1)=(H_γ,0,n)-m(n-1).
  
  
  
  
  
  
  
  
  
   §.§ Step 2
  
  As a consequence of the previous discussion and the fact that m general points on a smooth rational curve of degree γ lie in general position for m≤γ +1, we obtain:
  
  
  Let γ_1,γ_2≥ 1 and m ≤min{n+1,γ_2+1} be integers.
   The incidence variety  
  
      I_γ_1,γ_2,n^m = {(Γ_1,Γ_2) |Γ_1 ∩Γ_2 = {p_1,…,p_m}}⊆ H_γ_1,0,n× H_γ_2,0,n
  
  of two smooth rational curves of degree γ_1 and γ_2, intersecting transversely at m points in general position, is irreducible and unirational, of dimension 
  
      I_γ_1,γ_2,n^m  =  H_γ_1,0,n + m +  H_γ_2,0,n - m(n-1).
  
  
  
  By Lemma <ref> we have that the variety H_γ_2,0,n(p_1,…,p_m) is dominated by a variety P(p_1,…,p_m) isomorphic to a projective space, which parametrizes embeddings f:^1↪^n of degree γ_1 whose image contains the points p_i.
  We can construct a variety 𝒫 over H_γ_1,0,n× (^1)^m such that the fiber over a general point (C,p_1,…,p_m) with p_1,…,p_m∈ C is the projective space P(p_1,…,p_m). 𝒫 is generically a projective bundle over H_γ_1,0,n× (^1)^m (at least on the open subset where the projection map is proper and flat). Hence 𝒫 is irreducible and unirational. Moreover 𝒫 dominates I_γ_1,γ_2,n^m, by sending an embedding f in the fiber P(p_1,…,p_m) over (C,p_1,…,p_m) to the pair (f(^1),C). Thus I_γ_1,γ_2,n^m is irreducible and unirational as well, since asking the intersection to be transversal is an open condition.
  
  
  
  Let n,d≥ 3 and γ≥ 1 be integers.
   The incidence variety  
  
      I_d,γ,n = {(E,Γ) |   E∩Γ = {pt}}⊆ H_d,1,n× H_γ,0,n
  
  of an elliptic and a smooth rational curve intersecting transversely at one point is irreducible and unirational, of dimension 
  
      I_d,γ,n  =  H_d,1,n + 1 +  H_γ,0,n - (n-1).
  
  
  
  
  
  
  
   §.§ Step 3
  
  
  Recall that a K3 surface of degree 2n-2 with 3≤ n ≤ 5 is a complete intersection in ^n. More precisely, in the case n=3 the K3 surface is a quartic surface, while in the case n=4 (resp. n=5) the K3 surface is a complete intersection of a quadric and a cubic (resp. three quadrics). This is in fact the reason why we restrict to polarizations of degree ≤ 5.
  
  In particular the choice of a K3 surface X containing E∪Γ (or Γ_1 ∪Γ_2) is parametrized by an iterated Grassmannian G.  In the case n=3, G=|ℐ_E ∪Γ(4)|, in the case n=4, G=ℰ is a projective bundle over |ℐ_E ∪Γ(2)|, whose fiber over q ∈ |ℐ_E ∪Γ(2)| is ℰ_q defined in the exact sequence
  
      0 → H^0(𝒪_^4(1)) · q→ H^0(ℐ_E ∪Γ(3)) →ℰ_q → 0.
  
  Finally, in the case n=5, our parameter space is G=Gr(3,H^0(E∪Γ, ℐ_E∪Γ(2))). All these parameter spaces are rational. We are going to discuss in detail Case 2, as Case 1 can be handled analogously.
  
  Let k = d γ + d^2 - n + 1. We consider the following incidence variety
  
      I := {(X,E,Γ) |  E∪Γ⊆ X,  E ∩Γ ={pt}}⊆ G× I_d,γ,n.
  
  
  We denote by I^sm⊆ I the open subvariety of I containing triples (X,E,Γ) with X smooth.
  
   
  The image of π_1: I^sm→_2k sending a triple (X,E,Γ) to the isomorphism class of the smooth K3 surface X is open in _2k. In particular, since _2k is irreducible, either I^sm = ∅ or π_1 is dominant.
  
  
  Choose a basis {H,E,Γ} of U⊕⟨ -2k ⟩ such that the intersection matrix of {H,E,Γ} is as in <ref>, and let X be a U⊕⟨ -2k ⟩-polarized K3 surface. Up to the action of the Weyl group, we can assume that H ∈NS(X) is big and nef. By Saint-Donat's result <cit.>, H is very ample if and only if there is no element C∈NS(X) with C^2=-2 and HC=0, and no element D∈NS(X) with D^2=0 and HD=2. Both conditions are closed in _2k, and if H is indeed very ample, it embeds X in ^n as a surface containing the desired pair of curves.
  
  
  In order to show the unirationality of _2k, we will use the following proposition. We denote by π_2: I → I_d,γ,n the forgetful map that sends the triple (X,E,Γ) to the pair (E,Γ).
  
   
  Assume that there exists a pair (E,Γ) ∈ I_d,γ,n such that the fiber F:=π_2^-1(E,Γ) contains an element in I^sm. If the number I_d,γ,n+F coincides with the expected dimension of I
  
      _2k+PGL(n+1)+|E|+|Γ|=18+PGL(n+1),
  
  then π_2 is dominant. As a consequence I is unirational, π_1:I _2k is a dominant rational map, and thus, _2k is unirational.
  
  
  Since by assumption I^sm∅, then by Lemma <ref> the morphism π_1: I^sm→_2k is dominant and induces a dominant rational map π_1 : I _2k. The map π_1 can be decomposed as the composition q_2 ∘ q_1, where q_1 sends a triple (X,E,Γ) to the K3 surface X, and q_2 sends a (smooth) K3 surface to its isomorphism class. The set of automorphisms of ^n fixing a K3 surface X⊆^n is finite so that the dimension of the fiber of q_2 is equal to PGL(n+1). Moreover, the fiber of q_1 is 1-dimensional since the elliptic curve E moves in a 1-dimensional pencil and Γ is rigid on X. Since π_1 is dominant, necessarily I is at least the expected dimension
  
      _2k+PGL(n+1)+|E|+|Γ|=18+PGL(n+1).
  
  Now let d ≤F be the minimal dimension of the fibers of π_2. By semicontinuity of the fiber dimension, there exists on open dense subset U of I_d,γ,n such that π_2^-1(E',Γ')=d for all (E',Γ') ∈ U. If d < F, then
  
      I= I_d,γ,n + d < I_d,γ,n+F,
  
  which is a contradiction, since by assumption the number I_d,γ,n+F coincides with the expected dimension of I. This implies that π_2 is dominant and that the general fiber of π_2 is isomorphic to G, and thus it gives to I (generically) the structure of a projective bundle over I_d,γ,n (since by cohomology and base change π_2 is a projective bundle over the open subset of I_d,γ,n where π_2 is proper, flat and the fibers are isomorphic to G).
  In particular I is unirational, and therefore _2k is unirational as well.
  
  
  The previous proposition proves the existence of the diagram
  
  
      I @–>[dl]_π_1[dr]^π_2   
      ℳ_2k       I_d,γ,n
  
  whenever I^sm∅ and a certain equality of dimensions holds. As we already remarked, the general fiber of π_2 is isomorphic to the iterated Grassmannian G defined above, of dimension
  
      π_2^-1(E,Γ) =
      
       h^0(E∪Γ, ℐ_E∪Γ(4))-1     for  n=3, 
      
       h^0(E∪Γ, ℐ_E∪Γ(2)) - 1 + h^0(E∪Γ, ℐ_E∪Γ(3)) - 6     for  n=4, 
      
       3· (h^0(E∪Γ, ℐ_E∪Γ(2))-3)     for  n = 5.
  
  In the case n=4, we choose a quadric hypersurface through E∪Γ and then a cubic hypersurface through E∪Γ which is not a multiple of the chosen quadric.
  
  
  We check with Macaulay2 that there exists a pair (E,Γ) ∈ I_d,γ,n such that the fiber F=π_2^-1(E,Γ) contains one element in I^sm and that
  
      I_d,n,γ+F-PGL(n+1)=18.
  
  In order to compute the dimension of I_d,n,γ, we check with Macaulay2 that (E,Γ) is a smooth point of I_d,n,γ.
  Notice that the same strategy works analogously for Case 1, with the only difference that the previous number has to be 17 instead of 18. This follows from the fact that the two smooth rational curves are rigid on the K3 surface, and thus the fiber of q_1 in the proof of Proposition <ref> is 0-dimensional.
  All the experimental data can be found in Tables <ref>, <ref>, <ref>. The interested reader may find the Macaulay2 code at <cit.>.
  
  
  We are also adapting the above strategy to prove unirationality for some quasi-polarized K3 surfaces. We construct K3 surfaces having a node (i.e. having a unique A_1-singularity) and containing either a smooth rational curve or a smooth elliptic curve (see Section <ref>). 
  
  
  
   §.§ Search for the lattices
   
  
  In this section we explain how we obtain an exhaustive list of projective models of elliptic K3 surfaces of Picard number 3 in Case 1 and 2.
  
  In Case 2, the K3 surface is automatically elliptic. In Case 1, we actually have to check that the lattice contains a copy of the hyperbolic plane. By <cit.>, the lattices U⊕⟨ -2k⟩ are unique in their genus, so an even lattice L of rank 3, signature (1,2) and determinant 2k contains a copy of U if and only if the genus of L coincides with the genus of U⊕⟨ -2k ⟩  (for the definition of the genus of a lattice, we refer to <cit.>). This amounts to computing the discriminant group of L; we do not report these straightforward computations here. 
  
  In order to obtain an exhaustive, but finite, list of possible projective models for such K3 surfaces, we want to bound the degrees of the elliptic and smooth rational curves. This bound arises from the fact that curves of fixed genus and “high” degree do not lie on hypersurfaces of “small” degree. We explain this in detail for the case of two rational curves in ^3 meeting transversely at m points in general position; the strategy in the other cases will be completely analogous.
  
  Let Γ_1,Γ_2 be smooth rational curves in ^3 of degree γ_1 and γ_2, respectively, meeting at m points. Assuming the maximal rank conjecture for the nodal union of two curves (see <cit.> for a proof in the smooth case), the short exact sequence
  
      0 →ℐ_Γ_1 ∪Γ_2(4) →_^3(4) →_Γ_1 ∪Γ_2(4) → 0
  
  implies that h^0(ℐ_Γ_1 ∪Γ_2(4))>0 if and only if
  
      35=h^0(_^3(4)) > h^0(_Γ_1 ∪Γ_2(4))=4(γ_1+γ_2)+2-m.
  
  This allows us to obtain a (conjectural) bound on the degrees γ_1, γ_2 and on the number of intersection points m.
  Indeed notice that the number m of intersection points is bounded by the degrees γ_1,γ_2, since there are no smooth rational curves of degree γ passing through more than 2 γ points of ^n in general position (cf. the formula in Lemma <ref>). Therefore, the inequality <ref> provides a bound for the degrees γ_1,γ_2 of the two smooth rational curves. More precisely, if γ_1≥γ_2, then m≤ 2γ_2 and therefore
  
      35>4(γ_1+γ_2)+2-m≥ 4γ_1 + 2γ_1 + 2 ≥ 4γ_1+4,
  
  so γ_2≤γ_1≤ 7.
  
  
  Now, we can produce the list of all possible projective models of elliptic K3 surfaces of Picard number 3 in Case 1 and 2: for every γ_1≤γ_2 ≤ 7 and every m ≤ 2γ_1 satisfying the inequality (<ref>), we check whether the corresponding lattice contains a copy of the hyperbolic plane by looking at its genus.
  
  The same search works analogously in the case of nodal K3 surfaces since we only have to bound the degree of the smooth rational curve or of the smooth elliptic curve.
  
  
  
   §.§ Primitivity
   
  
  The K3 surfaces that we construct in Tables <ref>, <ref>, <ref> are in fact U⊕⟨  -2k ⟩-polarized K3 surfaces for suitable values of k.  In order to show this, we have to prove that the embeddings as in Equation <ref> are primitive. We will perform a case-by-case inspection, depending on the divisibility of k. Recall that, if D ∈ U⊕⟨ -2k ⟩ is divisible by r in NS(X), then r^2 | k.
  
  
  
   §.§ Table <ref>
  
  We consider the lattice embedding Λ^1_n,γ_1,γ_2↪NS(X) in Equation <ref>. Assume first k 50. We observe that k is either squarefree or a square, so we may assume that k is a square (i.e. k∈{25,36,49,64,81}), otherwise the embedding is automatically primitive. It is easy to notice that, if k=r^2, then Γ_1+Γ_2 is a vector of square 0 (since m=2 in all five cases) intersecting the three elements of the basis with multiplicity multiple of r. Therefore the embedding Λ^1_n,γ_1,γ_2↪NS(X) is primitive if and only if Γ_1+Γ_2 is not divisible in NS(X). Since Γ_1+Γ_2 represents the union of two rational curves on X meeting at two points, it is a fiber of an elliptic fibration on X, and therefore it is not divisible in NS(X).
  
  If instead k=50, the embedding is primitive if and only if the divisor H-Γ_2 is primitive in NS(X), since H-Γ_2 intersects the three elements of the basis with multiplicity 5. If by contradiction H-Γ_2 were divisible by 5, then 1/5(H-Γ_2) would have square 0 and intersection 1 with H; this is a contradiction since there are no curves of degree 1 and arithmetic genus 1.
  
  
  
   §.§ Table <ref>
  
  We consider the lattice embedding Λ^2_n,d,γ↪NS(X) in Equation <ref>. We are going to apply the following strategy for all the cases.
  
  Let D ∈Λ_n,d,γ^2 be a generator of ⟨ E,Γ⟩^⊥. Since ⟨ E,Γ⟩ is a copy of the hyperbolic plane, we have that D^2=-2k and the basis {E,Γ,D} gives an explicit isomorphism Λ_n,d,γ^2 ≅ U ⊕⟨ -2k ⟩. Therefore, the embedding Λ_n,d,γ^2 ↪NS(X) is primitive if and only if D is primitive in NS(X). Hence assume that 1/2D ∈NS(X) (thus 4 | k); if D is divisible by some other number r the argument is analogous. 
  A straightforward computation yields
  
      D=H-(γ+2d)E-dΓ.
  
  We distinguish some cases depending on the parity of d,γ.
   
  
  ∙  d≡γ≡ 0 2: In this case D is divisible by 2 if and only if H is divisible by 2, but the hyperplane section is primitive in NS(X).
  
  
  ∙ d≡ 1, γ≡ 0 2: D is divisible by 2 if and only if H-Γ is divisible by 2. But 1/2(H-Γ) would have square 0 and intersection 2 with H, and there are no curves of degree 2 and arithmetic genus 1.
  
  
  ∙ d≡ 0, γ≡ 1 2: D is divisible by 2 if and only if H-E is divisible by 2. If k=24, 1/2(H-E) would be a curve of degree 2 and arithmetic genus 1. If k=40, 1/2(H-E) would have square -2, so it would be either effective or anti-effective. But (H-E)H<0, (H-E)E>0, and this is a contradiction since H and E are nef. Finally, if k=68, 1/2(H-E) would have square -2 and intersection 0 with H; this is a contradiction since the K3 surfaces we are considering are generically smooth.
  
  
  ∙ d≡γ≡ 1 2: D is divisible by 2 if and only if H-E-Γ is divisible by 2. If k ∈{16,28}, 1/2(H-E-Γ) would have square -2, but (H-E-Γ)H<0 and (H-E-Γ)E>0 leads to a contradiction as above. If instead k=52, 1/2(H-E-Γ) would be a (-2)-curve orthogonal to H which is again a contradiction.
  
  
  
  
   §.§ Table <ref>
  
  The reasoning is completely analogous to the previous case.
  
  
  
  § EXPERIMENTAL DATA
  
  
  
  
   §.§ Case 1: Two smooth rational curves
  
  
  Let γ_1,γ_2 and m be integers. 
  Let Γ_1,Γ_2 ⊂^n be two smooth rational curves of degree γ_1 and γ_2, respectively, intersecting transversely at m points. If X⊆^n is a K3 surface containing the union Γ_1 ∪Γ_2, then there exists a lattice embedding
  
      Λ_n,γ_1,γ_2^1:=
      [ 2n-2  γ_1  γ_2;  γ_1   -2    m;  γ_2    m   -2 ]↪NS(X).
  
  For suitable choices, this lattice is isomorphic to U ⊕⟨ -2 k ⟩ for some integer k. In Table <ref> we specify this integer k in every case and list the data obtained with our Macaulay2 program.
  Note that there is more than one configuration of two smooth rational curves yielding a K3 surface in _2k. We only list one possibility for each k in Table <ref>.
  
  
  
   §.§ Case 2: An elliptic and a smooth rational curve
  
  
  Let d,n,γ be integers such that 3≤ d≤ 8, 3≤ n ≤ 5 and  1≤γ≤ 7. 
  Let E⊂^n be an elliptic curve of degree d, and let Γ⊂^n be a smooth rational curve of degree γ intersecting E transversely at one point. We denote by X⊆^n a K3 surface with the following lattice embedding 
  
      Λ_n,d,γ^2 :=
      [ 2n-2    d    γ;    d    0    1;    γ    1   -2 ]≅ U ⊕⟨ -2( d^2 + dγ + -n +1) ⟩↪NS(X).
  
  We set k = d^2 +d γ -n+1. In Table <ref> we specify this integer k in every case and list the data obtained with our Macaulay2 program. 
  
  k=10: 
  Let X⊆^3 be a smooth  quartic surface containing a line and an elliptic curve of degree 3 intersecting transversely at one point. Then X is a K3 surface with the following primitive lattice embedding
  
      [  4  3  1;  3  0  1;  1  1 -2 ]≅ U ⊕⟨ -20 ⟩↪NS(X).
  
  
  We recall the dimension count in this example:
  
      (Hilb_3· t(^3)) + 1 + (Hilb_t+1(^3)) - 2  + (h^0(E∪Γ, ℐ_E∪Γ(4))-1) - PGL(4)= 
          = 12 + 1 + (4-2) + (19-1) - (4^2-1) = 18  = _2k +  |E| +  |Γ|.
  
  
  
   
  
  
  
  
  
  
  
  
  § CONSTRUCTION OF NODAL ELLIPTIC K3 SURFACES
   
  
  
  We adapt our above construction in order to deal with nodal K3 surfaces having one A_1-singularity. This allows us to complete the proof of Theorem <ref>. Indeed, for certain values of k, the constructions explained above only yield examples where the locus I^sm, corresponding to smooth K3 surfaces, is empty. Therefore, allowing in this section the K3 surfaces to have a singular point, we are able to prove the unirationality of ℳ_2k for further values of k; notice that in all cases the general members of ℳ_2k will be the minimal resolutions of the nodal K3 surfaces constructed here.
  
  
  As in Step 1 of Construction <ref>, we construct a curve of degree d (that can either be a smooth rational or elliptic curve). In the second step we choose a point p∈^n. The final step is to construct a K3 surface X containing the given curve and having an A_1-singularity at p.  
  
  We restrict our considerations to the case of an elliptic curve E with a point p on it (the other cases are treated similarly). The desingularization of X contains the exceptional divisor C_p ≅^1 over the A_1-singularity.  Then we have the following lattice embedding (which will also be primitive by Section <ref>)
  
      [ 2n-2    0    d;    0   -2    1;    d    1    0 ]≅⟨ -2·(d^2 - n +1) ⟩⊕ U ↪NS(X).
  
  given by the intersection matrix with respect to the basis ⟨Ø_^n(1)|_X, C_p, E ⟩. 
  We set k'=d^2-n+1. Hence, we have an incidence variety 
  
      I' := {(X,E) | (E,p)∈ H^1_d,1,n and  E⊂ X ∈_2k'}⊂_2k'× H^1_d,1,n
  
  and the natural projections, denoted by π_1' and π_2'. The fiber π_2'^-1(E,p) is unirational. Indeed, to obtain a K3 surface that is nodal in a point, one has to solve linear equations in the coefficients of the equations generating the K3 surface (as well as their derivatives). But the choice is unirational, and therefore the incidence variety I' is unirational. The unirationality of _2k' follows by the construction of an example with the desired properties and a dimension count, as in Case 1 and 2 (see also Section <ref> for details). The following table lists our experimental data for such nodal K3 surfaces. We denote by π_2'^-1(E,p) the dimension of nodal K3 surfaces containing E and having a node at p∈ E.
  
   
  
  
  By studying nodal K3 surfaces containing a smooth rational curve with an A_1-singularity we can show the unirationality of _2k for the following values of k: 
  
      k∈{13, 17, 21, 25, 31, 37, 41, 61}.
  
  But we have already shown the unirationality of _2k for these values of k before (see Table <ref>). Therefore, we do not treat these cases in detail. 
  
  
  We end this section with three examples demonstrating how to choose a nodal K3 surface in ^n for n= 3,4 and 5 containing a given curve and having an A_1-singularity at a point. Furthermore, we recall the dimension count in these examples showing that the projection π_1':I' _2k' is dominant.
  
  Nodal quartic surfaces: k'=7.
  Given an elliptic curve E together with a point p in ^3, a nodal K3 surface containing E and with a node at p is a quartic generator of the ideal ℐ_Ep^2:= ℐ_E∩ (ℐ_p)^2.
  
  Let X'⊂^3 be a nodal quartic surface containing an elliptic curve E of degree 3 with an A_1-singularity at a point of E. Then the desingularization X of X' is a smooth K3 surface with the following primitive lattice embedding
  
      [  4  0  3;  0 -2  1;  3  1  0 ]≅⟨ -14 ⟩⊕ U ↪NS(X).
  
  We recall the dimension count in this case:
  
      (Hilb_3· t(^3)) + 1 + (h^0(ℐ_Ep^2(4))-1) - PGL(4)= 
          = 12 + 1 + (21-1) - (4^2-1) = 18  = _2k' + 1 .
  
  
  
  Nodal complete intersections of a quadric and a cubic: k' = 6.
  Given an elliptic curve E together with a point p, we get a nodal K3 surface by choosing two generators of degree 2 and 3 in the ideal ℐ_E with the same tangent space at the point p. Therefore, we compute all quadric and cubic hypersurfaces containing E and being tangent at p to a ^3 that contains the tangent line of E at p. The ideal of such hypersurfaces, denoted ℐ_E,^3, is the intersection of ℐ_E and ℐ^2_p∩ℐ_^3. 
  
  Let X'⊂^4 be a nodal K3 surface containing an elliptic curve E of degree 3 with an A_1-singularity at a point of E. Then the desingularization X of X' is a smooth K3 surface with the following primitive lattice embedding
  
      [  6  0  3;  0 -2  1;  3  1  0 ]≅⟨ -12 ⟩⊕ U ↪NS(X).
  
  We recall the dimension count in this case (cf. equation (<ref>)):
  
      (Hilb_3· t(^4)) + 1 +  (^3  containing the tangent line   T_p(E)) 
          + (h^0(ℐ_E,^3(2)) - 1 + h^0(ℐ_E,^3(3)) -6) - PGL(5)= 
          = 15 + 1 + 2 + (7 + 24 - 7) - (5^2-1) = 18  = _2k' + 1 .
   
  
  
  
  Nodal complete intersections of three quadrics: k'= 12.
  Given an elliptic curve E together with a point p, we obtain a nodal K3 surface by choosing a nodal quadric in the ideal ℐ_Ep^2:= ℐ_E∩ (ℐ_p)^2 and two further quadrics in the ideal of E.
  
  Let X'⊂^5 be a nodal K3 surface containing an elliptic curve E of degree 4 with an A_1-singularity at a point of E. Then the desingularization X of X' is a smooth K3 surface with the following primitive lattice embedding
  
      [  8  0  4;  0 -2  1;  4  1  0 ]≅⟨ -24 ⟩⊕ U ↪NS(X).
  
  We recall the dimension count in this case (cf. equation (<ref>), noticing that the -3 in the first line comes from the projectivization of H^0(ℐ_E(2))) and from the fact that the last two quadrics can be chosen up to multiples of the first quadric):
  
      (Hilb_4· t(^5)) + 1 + (h^0(ℐ_Ep^2(2))-1) + (2·(h^0(ℐ_E(2)) - 3)) - PGL(6)= 
          = 24 + 1 + (9-1) + (2·(13 - 3)) - (6^2-1) = 18  = _2k' + 1 .
  
  
  
  
  
  
  amsalpha
  
  
  
  