On K$K$ ‐stability of P3$\mathbb {P}^3$ blown up along the disjoint union of a twisted cubic curve and a line

We find all K$K$ ‐polystable smooth Fano threefolds that can be obtained as blowup of P3$\mathbb {P}^3$ along the disjoint union of a twisted cubic curve and a line.


Plan of the paper
In Section 2, we state the results that we will use to prove Main Theorem.In Section 3, we will discuss the equivariant geometry of ℙ 3 that will help us to understand the equivariant geometry of  in Family 3.12.We will focus our attention on the members in Family 3.12 for which the -polystability has not been proved yet.In this section, we show that Aut() ≅ Aut(ℙ 3 ,  + ) contains a subgroup  ≅ ℤ∕2ℤ × ℤ∕2ℤ.We will show that there are no -fixed points on ℙ 3 , and describe -invariant quadrics containing  on ℙ 3 and -invariant lines on ℙ 3 .At the end of this section, we give a description of the Mori cone and the cone of effective divisors on .Finally, in Section 4, we prove our Main Theorem.

Plan of the proof
If  is not -polystable, then it follows from [25,Corollary 4.14] that there exists a -invariant prime divisor  over  such that () ⩽ 0 where () was defined in [12], see also [3,Definition 1.18] and Section 2. Let  be the center of  on .Then,  is not a point since  has no -fixed points, and  is not a surface by [11,Theorem 10.1], so that  is a -invariant irreducible curve.
Then, we derive a contradiction as follows.
① We exclude the case when () is a line such that () ≠  and () ∩  = ∅.This is done in Lemma 4.1.② We exclude the case when () ⊂ .This is done in Lemma 4.2.③ We show that () is not contained in a -invariant quadric passing through .This is done in Lemma 4.3.④ Using ①, ②, ③, we deduce that () is not a line.⑤ It follows from () ⩽ 0 that  is contained in Nklt(, ) for some -invariant effective ℚdivisor  ∼ ℚ −  and  ∈ ℚ such that  < 3  4 .Moreover, it follows from ②, ③ and description of the cone of effective divisors on  that  is not contained in the surface in Nklt(, ).This is done in Corollary 4. 6. ⑥ Finally, we show that () should be a line in ℙ 3 , which contradicts ④.

PRELIMINARY RESULTS
Let  be a Fano variety with Kawamata log terminal singularities, let  be a reductive subgroup in Aut(), let  ∶ X →  be a -equivariant birational morphism, let  be a -invariant prime divisor in X, and let  = dim().
Definition 2.1.We say that  is a -invariant prime divisor over the Fano variety .If  is exceptional, we say that  is an exceptional -invariant prime divisor over .We will denote the subvariety () by   ().Let () =   () −   (), where   () is the log discrepancy of the divisor .
Theorem 2.3 [11,Theorem 10.1].Let  be any smooth Fano threefold that is not contained in the following 41 deformation families: №1.17, №2.23 where () is the positive part of the Zariski decomposition of the divisor −  − , and () is its negative part.

Basic properties of Fano threefolds in family №3.12
Let  be the smooth twisted cubic curve in ℙ 3 that is the image of the map ℙ 1 ↪ ℙ 3 given by let  be a line in ℙ 3 that is disjoint from , and let  ∶  → ℙ 3 be the blow up of ℙ 3 along  and .Then,  is a Fano threefold in family 3.12 and all threefolds in this family can be obtained this way.Note that there exists the following commutative diagram: where: •  is the blowup of a line , •  is the blowup of a curve , •  is the blowup of a curve  * , •  is the blowup of a curve  * , • the left dashed arrow is the linear projection from the line , • the right dashed arrow is given by the linear system of quadrics that contain , •  is a fibration into the del Pezzo surfaces of degree 6, •  is the contraction of the proper transforms of the quartic surface in ℙ 3 that is spanned by the secants of the curve  that intersect , • pr 1 and pr 2 are projections to the first and the second factors, respectively.
Let  be a plane in ℙ 3 ,   be the exceptional surface of  that is mapped to ,   be the exceptional surface of  that is mapped to , and  be -exceptional surface.Then  ∼ ℚ  * (4) − 2  −   , and −  ∼ ℚ  * (4) −   −   .

Consider the commutative diagram:
where  is a ℙ 1 -bundle given by the linear system | * (2) −   |,  is the blowup of  and the dashed arrow is given by the linear system of quadrics containing , and  is the blowup of  * .
which means that ( L) is a conic.The preimage of this conic on  is spanned by strict transforms of secants of  which intersect L. Therefore, () is spanned by secants of  that intersect .Note that the class of the preimage is Moreover, ( L) is a smooth conic and  is a ℙ 1 -bundle, and thus, the preimage of ( L) is a smooth surface, so it is smooth along L, and thus, the class of  in ℙ 3 is given by  ∼ ℚ  * (4) − 2  −   .

3.3.1
Types of threefolds in Family 3.12 We look at the projection from the line  that is disjoint from  to ℙ 1 : which gives a 3-cover of ℙ 1 : By Riemann-Hurwitz, we have that the degree of the ramification divisor is 4. The multiplicity in each ramification point is either 2 or 3, so we have three options: • there are two ramification points both of multiplicity 3, • there is one ramification point of multiplicity 3 and two ramification points of multiplicity 2, • there are four ramification points of multiplicity 2.
We see that there are at least two ramification points on .By acting on  by the PGL(2, ℂ), we can make these points to be  1 = [1 ∶ 0],  2 = [0 ∶ 1] on .Now we look at the line .It is the intersection of 2 planes that are tangent to  at points  1 and  2 (note that these planes are different because the plane intersects the cubic  in three points, so the same plane cannot be tangent to  at two points), so it is given by the equations: We have three cases: ①  1 =  2 = 0 so  is given by the equations: Here, we have two ramification points of multiplicity 3.This case was described in [3].The corresponding threefold  is -polystable in this case.
②  1 = 0,  2 ≠ 0 (which is symmetric to the case  1 ≠ 0,  2 = 0) so  is given by the equations: Using the action of ℂ * by the matrix that fixes : We can assume that  is given by Here, we have one ramification point of multiplicity 3 and two ramification points of multiplicity 2. This case was described in [3] where it was proven that  is not -polystable.
③  1 ≠ 0,  2 ≠ 0 so  is given by the equations Using the action of ℂ * by the matrix that fixes : , . We can assume that  is given by Note that: •  ≠ 0 since otherwise we are in case ①, •  ≠ ±1 since otherwise  intersects  that is prohibited, •  ≠ ±3 since otherwise there exists a plane containing  that is tangent to  with multiplicity 3 (it is a plane given by  3 + 3 3 + 3 1 +  0 = 0 in case  = −3 and a plane − 3 + 3 3 − 3 1 +  0 = 0 in case  = 3), so this case is projectively isomorphic to the case ②.
Now the involution on ℙ 3 given by fixes  an .We can construct a similar involution for any pair of four ramification points on  ≅ ℙ 1 .This gives the action of ℤ∕2ℤ × ℤ∕2ℤ.More precisely, this group is generated by the involutions viewed on ℙ 1 : where  is any root of the equation  4 − 10 2 + 9.The action on ℙ 3 is given by the map induced by

ℤ∕2ℤ × ℤ∕2ℤ-fixed points on 𝑋
From now on, we assume until the end of this section that we are in case ③ of the previous part and  = ℤ∕2ℤ × ℤ∕2ℤ.In particular, Aut() is finite (see [8]).
Proof.Note that  ↪ Aut() since  is a spatial curve.If there exists a -invariant plane Π, consider the intersection of Π with .There are three points in Π ∩  counted with multiplicities.Thus, since the order of  is 4, then there is a -fixed point on  ≅ ℙ 1 , which is a contradiction.□

Corollary 3.2.
There are no -fixed points on ℙ 3 .
Lemma 3.4.The linear system  is three-dimensional, it contains exactly three -invariant surfaces, and these surfaces are smooth.
Proof.Note that this statement does not depend on the equivariant choice of coordinates because PGL 2 (ℂ) contains a unique subgroup isomorphic to ℤ∕2ℤ × ℤ∕2ℤ up to conjugation.So, we can choose coordinates such that the generators of our group will look like: This gives us the action on ℙ 3 by: The linear system  is clearly three-dimensional.We can provide the equations for 3 -invariant quadrics containing : Note that ( 1 ,  2 ) acts on the equation of: •  1 by multiplying it by (1, −1), •  2 by multiplying it by (1, 1), •  3 by multiplying it by (−1, 1).
Thus, since the action is pairwise distinct and  is three-dimensional, there are exactly three -invariant quadrics which we listed above.Note that these quadrics are smooth.□ Now take a -invariant quadric  ∈  and look at the intersection of it with .Note that  ⊄  since  does not intersect .The intersection  ∩  cannot consist one point since we do not have -fixed points, thus  ∩  consists of two distinct points.These two points do not belong to the same curve of bidegree (1,0) or (0,1) (since these curves are the lines on  and we know that  ⊄ ).Now we see that the blowup Q →  at these points is a del Pezzo surface of degree 6.

ℤ∕2ℤ × ℤ∕2ℤ-invariant lines
Let us describe -invariant lines in ℙ 3 .Assume that  is generated by  1 and  2 , as in the proof of Lemma 3.4.In this case, all -invariant lines are of the form: where [ ∶ ] ∈ ℙ 1 .All such lines do not intersect each other and lie on the quadric  4 given by  1  0 =  2  3 .We see that ℙ 3 contains infinitely many -invariant lines and all of them are contained in  4 .Among them, there are three lines that intersect .We can describe them explicitly.The intersection of this quadric with  consists exactly of six points that are: here in the third column are given the corresponding coordinates on  ⊂ ℙ where the bottom dashed arrow is given by the linear system of quadrics containing .Using the equations of quadrics that form the basis of the linear system  defined in Section 3.3.3,we get the explicit map: We know that •() is a conic.Let us write its equation in ℙ 2 with coordinates [ ∶  ∶ ]:  1  2 +  2  +  3  +  4  2 +  5  +  6  2 = 0.
We want to look at the preimage of this equation in ℙ 3 that will give the equation for ().
into the defining equation of •(), we get: Recall from Section 3.3.4that all -invariant lines are of the form where for some  ∈ ℂ.Note that  ≠ 0 since otherwise  would have an infinite group of automorphisms.
Similarly, () is given by for  ∈ ℂ.By our assumption,   is contained in ().This gives So that () is given by Similarly, since   is contained in (), we get and the solution to this system is  = , which means that   =  and   = () coincide contradicting the assumption on .□

Mori cone
Let Proof.Suppose () ⊂ ℙ 3 is the surface of degree  in ℙ 3 .Then, we have where   is the multiplicity of () in  and   is the multiplicity of () in .Suppose that  ≠   ,  ≠   and  ≠  for all .Now let uss intersect  with three extreme rays   ,   ,   corresponding to , , : So, we have that: Moreover, if  1 is the general line intersecting  and  2 is the general secant of , then we get strict inequalities: Now we want to find the integer positive solutions for: Comparing the coefficients, we get: The nonnegative solution to this system can be given by

PROOF OF THE MAIN THEOREM
As described in Section 1.2, we will prove that  is -polystable.Let  =   () be a -invariant irreducible curve where  is a -invariant prime divisor  over  such that () =   () −   () ⩽ 0.
Lemma 4.1.Suppose that () ≠ , then () is not one of the -invariant lines that does not intersect .
Proof.Let us take a -invariant line () that does not intersect  and consider a general plane  that contains this line.It intersects a line  in one point and a twisted cubic  at three points.
Let  be the proper transform of  on .In this case, we have that the induced map |  ∶  →  is the blowup of a plane  in four points  1 =  ∩ ,  2 ,  3 ,  4 =  ∩ .We now need to check that these points are in general position to conclude that  is a del Pezzo surface of degree 5.
To prove that we need to show that the points in { 1 ,  2 ,  3 ,  4 } are in general position which means that no three of them belong to the same line.Note that  2 ,  3 ,  4 =  ∩  do not belong to the same line, because  is an intersection of quadrics.So, the only option is that  1 and two points from the set { 2 ,  3 ,  4 } belong to the same line.Suppose that  is a general plane and  1 and 2 points among { 2 ,  3 ,  4 } are contained in one line .From Section 3.2, we know that () is spanned by secants of  that intersect , so  contains such secant .Moreover, () intersects , so we see that () intersects a general secant of  that is contained in ().Then,  ⊂  that contradicts Lemma 3.5.So, we can choose the hyperplane  in such a way that the points in { 1 ,  2 ,  3 ,  4 } are in general position.Thus,  is a del Pezzo surface of degree 5. Let  1 ,  2 ,  3 ,  4 be the exceptional divisors corresponding to points  1 ,  2 ,  3 ,  4 , respectively, and   be the preimages of lines connecting   and   for  ∈ {1, … , 4}.Recall that  1 ,  2 ,  3 ,  4 , and   generate the Mori Cone NE().We have that −  ∼  * (4) −   −   ,  ∼  * (4) − 2  −   , and moreover, The intersections are given by:  2 , which implies that −  −  is not pseudoeffective for  > 3∕2.Let () = (−  − ) be a positive part of Zariski decomposition and () = (−  − ) be a negative part of Zariski decomposition.Here, we use the notations introduced in Theorem 2.4. and Then take any  ∈ ℝ ⩾0 .Suppose that (, ) is a positive part of the Zariski decomposition of (−  − )|  − , and (, ) is a negative part of the Zariski decomposition of (−  − )|  − .The intersections of (, ) with the generators of NE(S) above are: So, we obtain: The obtained contradiction completes the proof of the lemma.□ Lemma 4.2.One has  ⊄   .
1 Suppose that  ≠ |   and  ∼  + .Note that  ⩾ 1 since ℙ 3 does not contain -fixed points.Then using the convexity of volume we get the inequality, we obtain so it is enough to show that the last integral is less than 1 to deduce a contradiction.So, suppose  ∼ .We have that: The intersections of (, ) with the generators of NE() are: Hence, for any -invariant curve  ⊂   such that  ≠ |   , we have Corollary 2.4, we obtain the desired contradiction.
2 Suppose  = |   ∼  + 3.Take any  ∈ ℝ ⩾0 , then we have: The intersections of (, ) with the generators of NE() are: Hence, by Corollary 2.4, we obtain the desired contradiction that completes the proof of the lemma.□ Proof.Suppose that  ⊂ .Note that () is a -invariant quadric that contains .Recall that it is smooth.Let us seek for a contradiction.Let us identify () = ℙ 1 × ℙ 1 such that  is a curve in () of degree (1,2).Then,  induces a birational morphism  ∶  → ℙ 1 × ℙ 1 that is a blowup of two intersection points () ∩  = { 1 ,  2 }, which are not contained in the curve .Moreover, the surface  is a smooth del Pezzo surface of degree 6, because the points of the intersection () ∩  are not contained in one line in () since otherwise this line would be .But  is not contained in () that is a contradiction.By Theorem 2.3, we have which implies that −  −  is nef for every  ∈ [0, 1].On the other hand, we have . We denote () = (−  − ) and () = (−  − ).Then, we have and Suppose  ∶  → ℙ 1 × ℙ 1 is the blowup at points  1 ,  2 with  1 and  2 the exceptional curves of  that correspond to points  1 ,  2 , respectively.We denote by  1 and  2 the proper transforms on  of general curves  1 and  2 in ℙ 1 × ℙ 1 of degrees (1,0) and (0,1), respectively.The intersections are given by: We 1 Suppose  ≠   |  , then () is a curve since  ≠  1 and  ≠  2 , because neither  1 nor  2 is -invariant.Now we have that () ∼  1 +  2 and so where  1 is the multiplicity of () at point  1 and  2 is the multiplicity of () at point  2 .Note that  exchanges  1 and  2 and  is a -invariant curve, thus  1 =  2 =∶ .We know that  ∉ { 11 ,  12 ,  21 ,  22 } since the   are not -invariant for any , .Thus:

Now we have that
) and similarly, for  2 and  3 .So, it is enough to get Hence, by Corollary 2.4, we obtain the desired contradiction.
Case 2. Suppose  ∼  2 then the intersections of (, ) with the generators of NE() are given in the following table: So, we obtain: Hence, by Corollary 2.4, we obtain the desired contradiction.
1 −  2 to deduce a contradiction.Take any  ∈ ℝ ⩾0 .Suppose that (, ) and (, ) are positive and negative parts part of the Zariski decomposition of (−  − )|  − , respectively.Case 1. Suppose  ∼  1 then the intersections of (, ) with the generators of NE() above are given in the following table: