Minimal Terracini Loci in a Plane and Their Generalizations

: We study properties of the minimal Terracini loci, i

For all positive integers x and any variety X, let S(X, x) denote the set of all A ⊂ X such that #A = x.For any smooth point p of X, let (2p, X) (or just 2p if X = P n ) be the closed subscheme of X, with (I p ) 2 as its ideal sheaf.Hence, (2p, X) is a zero-dimensional scheme of degree dim X + 1 with {p} as its reduction.For any finite subset S of X contained in the smooth locus of X, set (2S, X) := ∪ p∈S 2p.If X = P n , set 2S := (2S, P n ).For any set A ⊂ P n , let ⟨A⟩ denote its linear span.Fix positive integers n, d and x.The Terracini locus T(n, d; x) is the set of all S ∈ S(P n , x) such that ⟨S⟩ = P n , h 0 (I 2S (d)) > 0 and h 1 (I 2S (d)) > 0 [12][13][14].More important is the minimal Terracini locus T(n, d; x) ′ , which is the set of all S ∈ T(n, d; x) such that h 1 (I 2A (d)) = 0 for all A ⊊ S.
To the best of our knowledge, the notion of minimality for Terracini sets was explicitly defined for Veronese embeddings in [13] and for arbitrary varieties in [12].Since it is a very natural notion, it occurs "in nature" even if it is not explicitly defined.For instance, in the list in [19] of cardinality 3 Terracini sets for the Segre embeddings, the non-minimal ones are [19], Examples 4.1 and 4.2.
The minimal Terracini locus is usually very different from the non-minimal one [12][13][14].In the setup of the Veronese embeddings on P n , the minimal one and the non-minimal one were considered in [13].In that paper, many differences were pointed out.For instance, for almost all pairs (n, d), we have T(n, d; x) ̸ = ∅ for all x ≫ 0 ( [13], Th. 1.1(iii)), while T(n, d; x) ′ = ∅ for all x > ⌈( n+d n )/(n + 1)⌉ ( [13], Prop.3.1).In this paper, we only consider the case n = 2 (as in [13]), and our tools (mainly the Hilbert function of the critical schemes of the elements of T(2, d; x) ′ ) are strong enough only for n = 2.For the case n > 2, we raise several questions.
We prove the following results.Theorem 2. Fix a positive integer e.Then there is an integer d(e) ≥ 3 such that for all integers d ≥ d(e), we have T(2, d; x) ′ ̸ = ∅ for e consecutive integers x.
Theorem 3 shows that for d ≫ 0, there are arbitrarily large consecutive gaps and arbitrarily large consecutive non-gaps.Question 2. Is it possible (taking a larger d 1 (e) or a larger d(e)) to get that there are exactly c consecutive gaps or non-gaps?
Our tools for making large consecutive gaps or large consecutive non-gaps seems not to be able to address Question 2.
As in [14], a key tool is the numerical character of any critical scheme of any S ∈ T(2, d; x) ′ (see Section 2 on the preliminaries).
In Section 4, we prove the results on the possible degrees of the critical schemes of S ∈ T(2, d; x) ′ (Theorem 6).In particular, we prove that Z ̸ = S (Proposition 1).Then, we prove the following theorem.Theorem 4. Fix an integer c ≥ 3, and set d 0 (c) := 6c + 3. Then for all d ≥ d 0 (c), there are integers x i , 1 ≤ i ≤ c with the following properties: 1.
There is S ∈ T(2, d; x i ) ′ with a critical scheme Z with deg(Z) = 2x i ; 3.
There is no positive integer y such that there is A ∈ T(2, d; y) ′ with a critical scheme Z ′ with Theorem 4 is analogous to [14], Th. 1.3 for the degrees of the critical schemes of minimally Terracini sets.
In Section 6, we consider several related definitions of Terracini sets.One of the main results (Theorem 7) applies also to the degrees of the critical schemes of elements of T(2, d; x) ′ .It says that for d ≫ 0, there are arbitrarily large gaps in the degrees of critical schemes.
In the last section, we discuss some questions related to the maximal integer x such that T(n, d; x) ′ ̸ = ∅.
It would be very interesting to extend [20,21] to some or all toric surfaces.Even an extension to only P 1 × P 1 would be nice.
We thank the referees for suggestions, which improved the presentation of the paper.

Preliminaries
We work over an algebraically closed field with characteristic 0. Each set T(n, d; x) and T(n, d; x) ′ is constructible ( [22], Ex.II.3.18,Ex.II.3.19), and hence, we may speak about the dimensions and the irreducible components of the Terracini loci and the minimal Terracini loci.
For any zero-dimensional scheme Z ⊂ P 2 , Z ̸ = ∅, let τ(Z) denote the maximal integer ≥ −1 such that h 1 I Z (d)) > 0. Let s(Z) be the first integer s such that h 0 (I Z (s)) > 0. The numerical character n 0 , . . ., n s−1 , s := s(Z) is a string of s integers n 0 ≥ n 1 ≥ • • • ≥ n s−1 that uniquely determines the Hilbert function of Z [14,20,21].We have n 0 = τ(Z) + 2 and n s−1 ≥ s.The numerical character n 0 , . . ., n s−1 is said to be connected if n i ≤ n i+1 + 1 for i = 0, . . ., s − 2. Fix any S ∈ T(n, d; x).A critical scheme of S is a subscheme Z ⊂ 2S such that each connected component of Z has a degree of at most 2. If S ∈ T(n, d; x) ′ , then Z red = S for all critical schemes Z of S ([13], Lemma 2.11).The numerical character of any critical scheme of any element of T(2, d; x) ′ is connected ( [14], Th. 2.10).
Remark 1. Fix integers d > t > 0. Let T ⊂ P 2 be any integral degree t curve.Since h 1 (O P 2 (d − t)) = 0, the long cohomology exact sequence associated with the inclusion D ⊂ P 2 gives h We use the following result ([14], Lemma 2.9).Lemma 1.Let Z ⊂ P 2 , Z ̸ = ∅ be a zero-dimensional scheme.Set z := deg(Z), s := s(Z) and d := τ(Z).Assume that the numerical character n 0 , . . ., n s−1 is connected, s ≤ (d + 3)/2, and there exists an integer t such that t 2 ≤ z and z t + t − 3 ≤ d.Then t = s, z = s(d + 3 − s) and Z is the complete intersection of a curve of degree z/t and a curve of degree t.
Proof of Theorem 1: By [13], Proposition 3.5, we have T(2, d; x) ′ = ∅ for all x > ρ.By [12], Th. 2, we have T(2, d; ρ) ′ ̸ = ∅ if d ≡ 1, 2 (mod 3), i.e., if ρ = (d + 2)(d + 1)/6.Now, assume d ≡ 0 (mod 3).For all integers x such that 0 ≤ x ≤ (d − 1)(d − 2)/2, let W(d, x) denote the Severi variety of all integral and nodal curves with exactly x nodes.The set W(d, x) is an irreducible variety of dimension ( d+2 2 ) − 1 − x [23,24,26,27].Take a general C ∈ W(d, ρ) and set S := Sing(C).Since S is contained in the singular locus of a degree d curve, Thus, to conclude the proof, it is sufficient to prove that S is minimal.Let E be the set of all subsets of S with cardinality ρ − 1.The semicontinuity theorem for cohomology gives that, restricting W(d, ρ) to an open dense subset W, we may assume that all Sing(D), D ∈ W have subsets of cardinality ρ − 1, B with the same h 1 (I 2B (d)) (so either all Sing(D) are minimal or none is minimal).We may assume C ∈ W. Fix A ∈ E and assume h 1 (I 2A (d)) > 0. Thus, h 0 (I 2A (d)) ≥ 2. Hence, there is a 1-dimensional family of curves with A contained in their singular locus.Since C is irreducible, the general element of this 1-dimensional family is irreducible.Varying D in W, we get a family W of integral degree d curves with at least ρ − 1 nodes, and dim W = dim V(d, ρ) + 1 = dim V(d, ρ − 1) and h 1 (I 2B (d)) > 0 for all B ∈ S(P 2 , ρ − 1) arising from some D ∈ W. The Severi conjecture proved in [23] also proves that each integral plane curve with at least ρ − 1 singular points is in the closure W(d, ρ − 1) of W(d, ρ − 1) (see the beginning of the Introduction of [26] or see [27] (in Italian) for a full proof).A general D ∈ W(d, ρ − 1) has as its singular locus a general element of S(P 2 , ρ − 1), and hence, h , which is a contradiction.

Remark 8.
As in [12], Th. 2, the proof of Theorem 1 gives the existence of an irreducible family of dimension 2ρ − 3 of the family of all S ∈ S(P 2 , ρ) formed by minimal Terracini sets.
Proof of Theorem 2: Set t := 4e + 4 and d(e) := 8t.Fix an integer d ≥ d(e).Note that d ≥ 8t.Since t ≡ 0 mod 4, the integer ( d+2 2 ) − ( d+2−t 2 ) = t(2d + 3 − t)/2 is even.Fix a general E ⊂ P 2 such that #E = 2e − 1. Remark 7 and the assumption on t give h 1 (I 2E (d)) = 0 and the existence of an integral and nodal degree t curve D such that Sing(D) = E. Take an odd integer a such that 1 ≤ a ≤ 2e − 1.Since a is odd, the integer gives the maximal possible number of independent conditions to the vector space H 0 (D, By Claim 1, we have h 0 (I B a (d − t)) > 0 and h 1 (I B a (d − t)) = 0. Thus, the long cohomology exact sequence of (2) gives h 0 (I 2S a (d)) > 0 and h 1 (I 2S a (d)) = 1.Hence, to conclude the proof of Claim 2, it is sufficient to prove that h In this case, we have the following residual exact sequence: the residual exact sequence of D gives the following exact sequence: Take an odd integer a such that 1 ≤ a ≤ 2e − 3. Thus, ), and A a+2 , f a+2 and B a+2 are well-defined.Since 3a Thus, taking all odd integers a between 1 and 2e − 1, we see that Claim 2 proves that T(2, d; x) ′ ̸ = ∅ for e consecutive integers.
Proof of Theorem 3: Set t := 2e + 4 and d 1 (e) := 8t.Note that t is even.Fix an integer d ≥ d 1 (e).We have d ≥ 8t.Set x := t(d + 3 − t)/2 and y := (t − 2)(d + 5 − t)/2.By [14], Proof of Prop.3.1, a general complete intersection of a curve of degree t/2 and a curve of degree d + 3 − t is an element of T(2, d; x) ′ , while a general complete intersection of a curve of degree (t − 2)/2 and a curve of degree d + 5 − t is an element of T(2, d; y) ′ .Since d ≥ 2t + e + 2, we have y < x − e. Fix an integer c such that 1 ≤ c ≤ x.Assume, by contradiction, the existence of S ∈ T(2, d; x − c) ′ , and let Z be a critical scheme of S. Set z := deg(Z).Since Z red = S ([13], Lemma 2.11) and each connected component of Z has a degree of at most 2, Since the numerical character of Z is connected ( [14], Th. 2.10), Claim 1 and Lemma 1 give c = 0, which is a contradiction.

Gaps for the Critical Schemes
In this section, we prove Theorem 4 and give several results on the degrees of critical schemes.Proposition 1.Take any S ∈ T(2, d; x) ′ , d ≥ 6 and any critical scheme Z of S. Then Z ̸ = S.
Theorem 5. Fix an integer t ≥ 4 and an integer d ≥ 3t such that d + 3 − t is even.Set x := t(d + 3 − t)/2.Then there is S ∈ T(2, d; x) ′ with a critical scheme of degree 2x.
Moreover, for all integers w such that there is no pair (y, A) such that A ∈ T(2, d; y) ′ and A has a critical scheme of degree w.
Proof.Let S ⊂ P 2 be a finite set that is the complete intersection of a smooth curve C of degree t and a curve of degree (d Since C is smooth, for any A ⊆ S, the residual exact sequence of C gives the following exact sequence: Thus, the long cohomology exact sequence of (7) shows that to prove that h 1 (I 2S (d)) = 1 and that h 1 (I 2A (d)) = 0 for all A ⊊ S (and hence, to prove that S is minimal), it is sufficient to prove that h 1 (I S (d − t)) = 0.This is true by [14], Proof of Prop. 3.
Example 1. Fix integers d ≥ r + 2 ≥ 5.There is a line L ⊂ P r and a smooth degree d nondegenerate rational curve X ⊂ P r such that X contains exactly 3 points of X and L is not a tangent line of X. Set S := L ∩ X. Obviously S ∈ T(X; 3).Since L is not one of the tangent lines of X, S is minimal.Obviously, S is the unique critical scheme of itself.
The following result is the equivalent of Theorem 3 for the degrees of the critical schemes: Theorem 7. Fix a positive integer e.Then there is a positive integer d 0 (e) such that for all d ≥ d 0 (e), there are integers 0 < x 1 < x 2 such that T(2, d; x i ) ′ ̸ = ∅, i = 1, 2, and there are S i ∈ T(2, d; x i ) ′ and critical schemes Z i of S i with deg(Z i ) = 2x i , while there is no (y, A, Z) with y a positive integer.A ∈ T(2, d; y) ′ , and Z is a critical scheme of A; hence, 2x 2 − e ≤ deg(Z) < 2x 2 .
Set t := 4e + 4 and d(e) := 8t.Fix an integer d ≥ d(e).Mimic the proof of Theorem 3 with S 2 as a complete intersection or apply Theorem 6.

Classification for d ≤ 8
In this section, we consider pairs (d, x) such that T(  (i) An element S ∈ S(P 2 , 8) is contained in T(2, 7; 8) ′ if and only if S is contained in a reduced conic D, with the restriction that if D is reducible, each irreducible component of D contains exactly 4 points of S.
Proof.Fix S ∈ T(2, 7; x) ′ , and call Z a critical scheme of S. Thus, z := deg(Z) ≤ 2x.By Remark 6, we have 8 ≤ x ≤ 12. Remark 6 also gives part (i).Thus, from now on, we assume 9 ≤ x ≤ 12. Since S is minimal, #(S ∩ L) ≤ 4 for all lines L and #(S ∩ D) ≤ 6 for any reduced conic D. Set s := s(Z).Recall that the numerical character n 0 , . . ., n s−1 ≥ s of Z is connected, n 1 < n 0 and n 0 = d + 2. For any plane cubic C, we have the following residual exact sequence: For any plane conic D, we have the following residual exact sequence: ) = 0).Thus, the long cohomology exact sequence of ( 12) (respectively, ( (b1.1) Assume the existence of the line L. We get h 1 (I Res L (Z) (6)) > 0. Since deg(Res L (Z)) ≤ 14, either there is a line R such that deg(R ∩ Res L (Z)) ≥ 8 or there is a conic T such that deg(T ∩ Res L (Z)) = 14.The conic T does not exist because it would contain at least 7 points of S. The line R does not exist because the reducible conic L ∪ R would contain at least 7 points of S.
(b1.2) Now assume the existence of the conic D. We have h 1 (I Res D (Z) (5)) > 0 with deg(Res D (Z)) ≤ 10.Thus, there is a line J such that deg(J ∩ Res D (Z)) ≥ 7, and hence, #(J ∩ S) ≥ 4. The theorem of Bezout gives J ⊂ C. Since deg(J ∩ Res D (Z)) ≥ 7, we get C = J ∪ D. We use the proof of step (b1.1) with J instead of L.
(b2) Now assume Z ⊂ C. First, assume that C is irreducible.Since C has arithmetic genus 1, h 1 (C, F ) = 0 for every rank 1 torsion free sheaf F on C. Since I Z,C (7) is a rank 1 torsion free sheaf on C with positive degree, we get a contradiction.Now assume that C is reduced.Since #(S ∩ L) ≤ 4 for all lines L and #(S ∩ D) ≤ 6 for any reduced conic D, we have C = L ∪ D, with L a line and D a reduced conic; # We conclude as in step (b1).

Generalized Terracini Loci
Definition 1. Fix a positive integer d and a zero-dimensional scheme W ⊂ P 2 such that h Definition 1 is a key definition because if Z is as in Definition 1 and A is any zerodimensional scheme containing Z, then h 1 (I A (d)) > 0, and hence the zero-dimensional schemes W such that h 1 (I W (d)) > 0 are, roughly speaking, built from its critical schemes.The next result, Theorem 8, says that each W such that h 1 (I W (d)) > 0 has a critical scheme.There are schemes W with several critical schemes (for instance the scheme 2S in [13], Th.Proof.Let E be the set of all A ⊆ W such that h 1 (I A (d)) ̸ = 0. Since W ∈ E, E ̸ = ∅.Take Z ∈ E with minimal degree such that h 1 (I Z (d)) > 0. The assumption on the minimality of deg(Z) implies h 1 (I Z ′ (d)) = 0 for all Z ′ ⊊ Z.Thus, Z is critical for W in degree d.
Let Z ⊆ W be any critical scheme of W in degree d.Since Z has subschemes of degree deg(Z) − 1 and h 1 (I A (d)) − h 1 (I B (d)) ≤ deg(B) − deg(A) for all zero-dimensional schemes A ⊂ B, we have h 1 (I Z (d)) = 1, and hence, h 1 (I Z (t)) = 0 for all t > d.Set s := s(Z) and z := deg(Z).Let n 0 , . . ., n s−1 be the numerical character of Z. Assume that n 0 , . . . ,n s−1 is not connected and let t be the first integer < s such that n t ≤ n t−1 − 2. By [21], Cor.3.2 there is a degree t curve T such that the scheme T ∩ Z has numerical character n 0 , . . . ,n t−1 (which is connected).Since n 0 = d + 2, h 1 (I Z∩T (d)) > 0. The minimality of Z gives Z = T ∩ Z.By the definition of s(Z), we get s = t, which is a contradiction.
In the next example, we give a list of possible connected components of zero-dimensional schemes A ⊂ P 2 that may be connected components of zero-dimensional schemes to which the easy Theorem 8 may be applied.It is important to notice that for interesting schemes W, the connected components may be completely different and with different degrees.
Example 2. For any positive integer m and any p ∈ P 2 , let mp denote the closed subscheme of P 2 with (I p ) m as its ideal sheaf.We have (mp) red = {p}, deg(mp) = ( m+22 ) and mp ⊂ (m + 1)p.We have 1p = {p}.By the Terracini Lemma ([1], Cor.1.11), the double point 2p is the scheme used to define the Terracini loci.It is easy to check that 2p is a flat limit of sets of cardinality 3 and that 3p is flat limit of 2p and a family of sets of cardinality 3. The scheme 4p is a flat limit of a family of union of 5 disjoint double points ([31], part 1 of Prop.22).Degree 5 subschemes of 3p containing 2p were used to compute secant varieties of tangential varieties of P 2 [32].General unions of schemes 4p (or its higher dimensional generalization) and double points are used to compute the dimension of the secant varieties of many varieties.Set Z (2; 0) = ∅.For each positive integer x and any p ∈ P 2 , let Z (x; p) denote the set of all curvilinear schemes Z ⊂ P 2 such that deg(Z) = x and Z red = {p}.Note that we require that every Z ∈ Z (x; p) to be curvilinear.The curvilinearity assumption is automatic for x = 1, 2, but it is a restriction for x > 2. The set Z (x; p) has a natural structure of a smooth and connected quasi-projective variety of dimension x − 1 [33][34][35][36].Since each A ∈ Z (x; p) is connected and curvilinear, for each integer 0 ≤ y ≤ x, there is a unique A ∈ Z (y; p) such that A ⊆ Z. Set Z (x) := ∪ p∈P 2 Z (x; p).The set Z (x) is a connected and smooth quasiprojective manifold of dimension x + 1.For any positive integer x and all e 1 , . . ., e x ∈ N, let Z (x; e 1 , . . ., e x ) denote the set of all (A, Z 1 , . . ., Z x ) such that A = (p 1 , . . ., p x ) ∈ S(P 2 , x) and Z i ∈ Z (e i ; p i ).For any (A, Z 1 , . . ., Z x ) ∈ Z (x; e 1 , . . ., e x ), set u(A, Z 1 , . . ., Z x ) := Z 1 ∪ • • • ∪ Z x ⊂ P 2 .The scheme u(A, Z 1 , . . . ,Z x ) is a degree e 1 + • • • + e x curvilinear scheme with exactly x connected components.Let u(Z; x; e 1 , . . ., e x ) denote the set of all u(A, Z 1 , . . ., Z x ) for some (A, Z 1 , . . ., Z x ) ∈ Z (x; e 1 , . . ., e x ).For all positive integers d, x, e 1 , . . ., e x , let T(d; x; e 1 , . . ., e x ) denote the set of all Z ∈ u(A, Z 1 , . . ., Z x ) such that h 1 (I Z (d)) > 0 and h 0 (I Z (d)) > 0. Take Z ∈ T (d; x; e 1 , . . ., e x ).We say that Z is minimal if h 1 (I W (d)) = 0 for all W ⊊ Z.Let T (d; x; e 1 , . . ., e x ) (respectively, T (d; x; e 1 , . . ., e x ) ′ ) denote the set of all Z ∈ T (d; x; e 1 , . . ., e x ) ′ (respectively, Z ∈ T (d; x; e 1 , . . ., e x ) ′ ) such that ⟨Z⟩ = P 2 .

Methods
There are no experimental data and no part of a proof is completed numerically.All results are given with full proofs.

Conclusions
We study properties of the minimal Terracini loci, i.e., families of certain zero-dimensional schemes, in the projective plane.Among the new results here are: a maximality theorem and the existence of arbitrarily large gaps or non-gaps for the integers x for which the minimal Terracini locus in degree d is non-empty.We study similar theorems for the critical schemes of the minimal Terracini sets.
We consider more general zero-dimensional schemes and give five open questions.Most of these question concern the extension of this paper to higher-dimensional projective space.
A different (and much more general) kind of extension would be to toric varieties.Even just for smooth toric surfaces, an extension should come with very nice examples and, for low cardinality sets, a full classification list.F. Galuppi, P. Santarsiero, D.A. Torrance and E. Teixeira Turatti studied in several (non-toric) cases the first non-empty Terracini locus [17].In particular, they gave a full classification for all smooth Del Pezzo surfaces.All elements of the first non-empty Terracini set are minimal.In those cases (and in particular for Del Pezzo surfaces and for the Hirzebruch surfaces), two natural questions arise: 1.
Are non-minimal Terracini loci non-empty for all numbers x ≫ 0? 2.
What is the computation of the cardinality of the second non-empty Terracini locus?
For (1), there should be finitely many classes of exceptional cases, i.e., of pairs (variety, embedding) in which all Terracini loci are empty and "almost all" the other pairs should have non-minimal Terracini sets for all x ≫ 0. These statements are known in the case of Veronese embeddings [13].

1 . 4 Theorem 8 .
for odd values of d).Fix a positive integer d and a zero-dimensional scheme W ⊂ P 2 such that h 1 (I W (d)) > 0. (a) W has at least one critical subscheme in degree d.(b) Let Z be any critical subscheme of Z in degree d.Then h 1 (I Z (d)) = 1, τ(Z) = d and the numerical character of Z is connected.
2, and the restriction map H 0 (O P 2 (d)) → H 0 (T, O T (d)) is surjective.Take any S ∈ T(2, d; x) ′ and any critical scheme Z of S. By ([13], Lemma 2.11), we have Z red = S, and hence, deg(Z) ≥ x.Easy examples show that the latter inequality is not true (for many d and x) for the critical schemes of elements of T(2, d; x) that are not minimal.