Abstract
Verlinde bundles are vector bundles \(V_k\) arising as the direct image \(\pi _*(\mathscr {L}^{\otimes k})\) of polarizations of a proper family of schemes \(\pi :\mathfrak X \rightarrow S\). We study the splitting behavior of Verlinde bundles in the case where \(\pi \) is the universal family \(\mathfrak X \rightarrow \left{\mathscr {O}(d)}\right\) of hypersurfaces of degree d in \(\left{\mathscr {O}(d)}\right\) and calculate the cohomology class of the locus of jumping lines of the Verlinde bundles \(V_{d+1}\) in the cases \(n=2,3\).
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1 Introduction
Let \(\pi :\mathfrak X \rightarrow S\) be a proper family of schemes with a polarization \(\mathscr {L}\). For \(k\ge 1\), if the sheaf \(\pi _*(\mathscr {L}^{\otimes k})\) is locally free, we call it the kth Verlinde bundle of the family \(\pi \).
For example (Iyer 2013), let \(C\rightarrow T\) be a smooth projective family of curves of fixed genus. Consider the relative moduli space \(\pi :{\text {SU}}(r)\rightarrow T\) of semistable vector bundles of rank r and trivial determinant. This family is equipped with a polarization \(\varTheta \), the determinant bundle. The Verlinde bundles \(\pi _*(\varTheta ^k)\) of this family are projectively flat (Hitchin 1990; Axelrod et al. 1991), and their rank is given by the Verlinde formula.
In this article, we study the example of the universal family \(\pi :\mathfrak {X} \rightarrow \left\mathscr {O}_{\mathbb {P}^n}(d)\right\) of hypersurfaces of degree d in the complex projective space \(\mathbb {P}^n\), with \(n>1\). This family comes equipped with the polarization \(\mathscr {L}\) given by the pullback of \(\mathscr {O}(1)\) along the projection map \(\mathfrak {X} \rightarrow \mathbb {P}^n\). For \(k\ge 1\), the sheaf \(\pi _{*}\mathscr {L}^{\otimes k}\) is locally free, as can be seen by considering the structure sequence of an arbitrary hypersurface of degree d in \(\mathbb {P}^n\). For \(k\ge 1\), we denote the kth Verlinde bundle of the family \(\pi \) by \(V_k\).
To better understand \(V_k\) we study its splitting type when restricted to lines in \(\left{\mathscr {O}(d)}\right\).
Let \(T\subseteq \left{\mathscr {O}(d)}\right\) be a line. On \(T=\mathbb {P}^1\), we define the vector bundle \(V_{k,T}:=V_k_T\). The splitting type of \(V_{k,T}\) is the unique nonincreasing tuple \((b_1,\ldots , b_{r^{(k)}})\) of size \(r^{(k)}:={\text {rk}}V_k\) such that \(V_{k,T} \simeq \bigoplus _i \mathscr {O}(b_i)\).
Sequence (2.1) puts constraints on the \(b_i\): they are all nonnegative and they sum up to \(d^{(k)}:=\deg (V_k)\). The set of such tuples \((b_i)\) can be ordered by defining the expression \((b'_i) \ge (b_i)\) to mean
With this definition, smaller types are more general: the vector bundle \(\mathscr {O} (b_i)\) on \(\mathbb {P}^1\) specializes to \(\mathscr {O} (b'_i)\) in the sense of Shatz (1976) if and only if \((b'_i) \ge (b_i).\)
If \(d^{(k)} \le r^{(k)}\), then the most generic possible type has thus the form \((1,\ldots ,1,0,\ldots ,0)\). We call this the generic splitting type. A computation shows that \(d^{(k)} \le r^{(k)}\) if \(k\le 2d\).
We have the following result on the cohomology class of the set of jumping lines
in the Grassmannian of lines in \(\left{\mathscr {O}(d)}\right\):
Theorem 1
Let \(n \le 3\), let Z be set of jumping lines of \(V_{d+1}\), and let [Z] be the class of Z in the Chow ring \({\text {CH}}(\mathbb G \mathrm r(1,\left{\mathscr {O}(d)}\right))\). We have
Furthermore, let b range over the integers with the property \(0\le b < \frac{\dim Z}{2}\) and define \(a=\dim Z  b, a'=a+\frac{{{\mathrm{codim}}}Z\dim Z}{2}\), \(b'=b+\frac{{{\mathrm{codim}}}Z\dim Z}{2}\).

1.
If \(\dim Z\) is odd or \(n=2\), we have
$$\begin{aligned}{}[Z] = \sum _{a,b} \left( {\left( {\begin{array}{c}a+1\\ n\end{array}}\right) }{\left( {\begin{array}{c}b+1\\ n\end{array}}\right) }{\left( {\begin{array}{c}a+2\\ n\end{array}}\right) }{\left( {\begin{array}{c}b\\ n\end{array}}\right) }\right) \sigma _{a',b'}. \end{aligned}$$(1.1) 
2.
If \(\dim Z\) is even and \(n=3\), we have
$$\begin{aligned}{}[Z]&= \sum _{a,b} \left( {\left( {\begin{array}{c}a+1\\ n\end{array}}\right) }{\left( {\begin{array}{c}b+1\\ n\end{array}}\right) }{\left( {\begin{array}{c}a+2\\ n\end{array}}\right) }{\left( {\begin{array}{c}b\\ n\end{array}}\right) }\right) \sigma _{a',b'} \nonumber \\&\quad + \left( {\begin{array}{c}\frac{\dim Z}{2} + 2\\ n\end{array}}\right) \left( {\begin{array}{c}\frac{\dim Z}{2}\\ n\end{array}}\right) \sigma _{\frac{\dim Z}{2},\frac{\dim Z}{2}}. \end{aligned}$$
The computation is carried out by the method of undetermined coefficients, leading into various calculations in the Chow ring of the Grassmannian. The assumption \(n\le 3\) is needed for a certain dimension estimation.
2 Attained splitting types
There exists a short exact sequence of vector bundles on \(\left{\mathscr {O}(d)}\right\)
as can be seen by taking the pushforward of a twist of the structure sequence of \(\mathfrak X\) on \(\mathbb {P}^n \times \left{\mathscr {O}(d)}\right\). The map M is given by multiplication by the section
In particular, we have \(r^{(k)} = \left( {\begin{array}{c}k+n\\ n\end{array}}\right)  \left( {\begin{array}{c}k+nd\\ n\end{array}}\right) \) and \(d^{(k)}=\left( {\begin{array}{c}k+nd\\ n\end{array}}\right) \).
Lemma 1
Let \(\mathscr {E}\) be a free \(\mathscr {O}_{\mathbb {P}^1}\)module of finite rank, and let
be a short exact sequence of \(\mathscr {O}_{\mathbb {P}^1}\)modules. Given a splitting \(\mathscr {E}'' = \mathscr {E}''_1 \oplus \mathscr {O}\), we may construct a splitting \(\mathscr {E} = \mathscr {E}_1 \oplus \mathscr {O}\) such that the image of \(\varphi \) is contained in \(\mathscr {E}_1\).
Proof
Define \(\mathscr {E}_1 :=\ker (\mathrm {pr}_2\circ \psi )\), which is a locally free sheaf on \(\mathbb {P}^1\). By comparing determinants in the short exact sequence \(0 \rightarrow \mathscr {E}_1 \rightarrow \mathscr {E} \rightarrow \mathscr {O} \rightarrow 0\) we see that \(\mathscr {E}_1\) is free, hence by an \({\text {Ext}}^1\) computation the sequence splits. The property \({\text {im}}(\varphi ) \subseteq \mathscr {E}_1\) follows from the definition. \(\square \)
Proposition 1
Let \(f_1, f_2 \in \left{\mathscr {O}(d)}\right\) span the line \(T \subseteq \left{\mathscr {O}(d)}\right\) and let p be the number of zero entries in the splitting type of \(V_{k,T}\). We have
Proof
Note that the map \(M_T\) sends a local section \(\xi \otimes \theta \) to \(s\xi \otimes f_1 \theta + t\xi \otimes f_2 \theta \). In particular, the image of \(\mathscr {O}(1)\otimes U\) is contained in \(\mathscr {O} \otimes (f_1 U + f_2 U)\). It follows that \(p \ge \dim H^0(\mathbb {P}^n,\mathscr {O}(k))  \dim ({f_1 U + f_2 U})\).
To prove the other inequality, consider the induced sequence
and assume for a contradiction that \(\mathscr {E}'' \simeq \mathscr {E}_1''\oplus \mathscr {O}.\) By Lemma 1, we have a splitting \(\mathscr {O}\otimes (f_1 U + f_2 U) \simeq \mathscr {E}_1 \oplus \mathscr {O}\) such that \({\text {im}}(M_T) \subseteq \mathscr {E}_1\).
Consider the map \({\widetilde{M}}_T :(\mathscr {O} \otimes U) \oplus (\mathscr {O} \otimes U) \rightarrow \mathscr {O} \otimes (f_1 U + f_2 U)\) defined by
We obtain the matrix description of \({\widetilde{M}}_T\) from the matrix description of \(M_T\) as follows. If \(M_T\) is represented by the matrix A with coefficients \(A_{i,j} = \lambda _{i,j} s + \mu _{i,j} t\), then \({\widetilde{M}}_T\) is represented by a block matrix
with \(A'_{i,j} = \lambda _{i,j}\) and \(A''_{i,j} = \mu _{i,j}\).
The property \({\text {im}}(M_T)\subseteq \mathscr {E}_1\) implies that after some row operations, the matrix A has a zero row. By the construction of \({\widetilde{M}}_T\), the same row operations lead to the matrix B having a zero row, but this is a contradiction, since the map \({\widetilde{M}}_T\) is surjective. \(\square \)
Corollary 1
Let \(T \subseteq \left\mathscr {O} (d)\right\) be a line spanned by the polynomials \(f_1,f_2\). Assume that \(d^{(k)} \le r^{(k)}\). Let \(\theta \) range over a monomial basis of \(H^0(\mathbb {P}^n, \mathscr {O}(kd))\). The bundle \(V_{k,T}\) has the generic splitting type if and only if \(\langle f_1\theta ,f_2\theta \mid \theta \rangle \) is a linearly independent set in \(H^0(\mathbb {P}^n,\mathscr {O}(k))\). \(\square \)
Corollary 2
Let \(T \subseteq \left\mathscr {O} (d)\right\) be a line spanned by the polynomials \(f_1,f_2\), and let \(d^{(k)} \le r^{(k)}\). The bundle \(V_{k,T}\) has not the generic type if and only if \(\deg (\gcd (f_1,f_2)) \ge 2dk\). In particular, if \(d^{(k)} \le r^{(k)}\) but \(k>2d\) then the generic type never occurs.
Proof
By Corollary 1, the bundle \(V_{k,t}\) has nongeneric type if and only if there exist linearly independent \(g_1,g_2\in H^0(\mathbb {P}^n,\mathscr {O}(kd))\) such that \(g_1f_1+g_2f_2 = 0\). Let \(h :=\gcd (f_1,f_2)\) and \(d':=\deg h\).
If \(d' \ge 2dk\) then \(\deg (f_i/h) \le kd\) and we may take \(g_1,g_2\) to be multiples of \(f_1/h\) and \(f_2/h\), respectively.
On the other hand, given such \(g_1\) and \(g_2\), we have \(f_1\mid g_2 f_2\), which implies \(f_1/h \mid g_2\), hence \(dd'\le kd\). \(\square \)
Proposition 2
Let \(k=d+1\). No types of \(V_k\) other than \((1,\ldots ,1,0,\ldots ,0)\) and \((2,1,\ldots ,1, 0,\ldots ,0)\) occur.
Proof
Assume that the type of \(V_k\) at some line \((f_1,f_2)\) is other than the two above. Then the type has at least two more zero entries than the general type. By Proposition 1, we have \(\dim \langle f_1 \theta , f_2 \theta \mid \theta \rangle \le 2d^{(k)}2\), so we find \(g_1,g_2,g'_1,g'_2\in H^0(\mathbb {P}^n,\mathscr {O}(1))\) and two linearly independent equations
with both sets \((g_1,g_2), (g'_1,g'_2)\) linearly independent. From the first equation it follows that \(f_1 = g_2 h\) and \(f_2 = g_1 h\), for some common factor h. Applying this to the second equation, we find \(g'_1 g_2 = g'_2 g_1\), hence \(g'_1 = \alpha g_1\) and \(g'_2 = \alpha g_2\) for some scalar \(\alpha \), a contradiction. \(\square \)
Corollary 3
Let \(k=d+1,\) let \(T\subset \left\mathscr {O} (d)\right\) be a line spanned by \(f_1,f_2\). The type \((2,1,\ldots ,1,0,\ldots ,0)\) occurs if and only if \(\deg (\gcd (f_1,f_2) \ge d1\). \(\square \)
3 The cohomology class of the set of jumping lines
Definition 1
Let \(k \ge 1\) and \((b_i)\) be a splitting type for \(V_k\). We define the set \(Z_{(b_i)}\) of all points \(t\in \mathbb G \mathrm r(1,\left{\mathscr {O}(d)}\right)\) such that \(V_{k,t}\) has splitting type \((b_i)\). For the set of points t where \(V_{k,t}\) has generic splitting type, we also write \(Z_{\text {gen}}\), and define the set of jumping lines\(Z:=\mathbb G \mathrm r(1,\left{\mathscr {O}(d)}\right) {\setminus } Z_{gen}\).
Now let \(k=d+1\). By Corollary 3, Z is the subvariety given as the image of the finite, generically injective multiplication map
sending the tuple \(((sg_1+tg_2)_{(s:t)\in \mathbb {P}^1},h)\) to the line \((shg_1+thg_2)_{(s:t)\in \mathbb {P}^1}\).
To perform calculations in the Chow ring A of \(\mathbb G \mathrm r(1,\left{\mathscr {O}(d)}\right)\), we follow the conventions found in Eisenbud and Harris (2016). We assume \({\text {char}}(k) = 0\) for simplicity. Let \(N:=\dim H^0(\mathscr {O}(d)) = \left( {\begin{array}{c}n+d\\ n\end{array}}\right) \). For \(N2\ge a\ge b\), we have the Schubert cycle
where \((H\subset H')\) is a general flag of linear subspaces of dimension \(Na2\) resp. \(Nb1\) in the projective space \(\left{\mathscr {O}(d)}\right\). The ring A is generated by the Schubert classes \(\sigma _{a,b}\) of the cycles \(\varSigma _{a,b}\). The class \(\varSigma _{a,b}\) has codimension \(a+b\), and we use the convention \(\sigma _{a}:=\sigma _{a,0}\).
Proof
(of Theorem 1) We have \(\dim Z = n+1+\left( {\begin{array}{c}d1+n\\ n\end{array}}\right) \) since Z is the image of the generically injective map \(\varphi \).
Let \(Q \subset \left{\mathscr {O}(d)}\right\) be the image of the multiplication map
The map f is birational on its image, since a general point of Q has the form gh with h irreducible. The Chow group \(A^{{{\mathrm{codim}}}Z}\) is generated by the classes \(\sigma _{a',b'}\) with \(N2\ge a'\ge b' \ge \left\lfloor \frac{{{\mathrm{codim}}}Z}{2}\right\rfloor \) and \(a'+b'={{\mathrm{codim}}}Z\), while the complementary group \(A^{\dim Z}\) is generated by the classes \(\sigma _{\dim Zb,b}\) with \(b\in {0,\ldots ,\left\lfloor \frac{\dim Z}{2}\right\rfloor }\). Write
We have \(\sigma _{a',b'} \sigma _{a,b} = 1\) if \(b'b = \left\lfloor \frac{{{\mathrm{codim}}}Z}{2}\right\rfloor \) and 0 else. Hence, multiplying the above equation with the complementary classes \(\sigma _{a,b}\) and taking degrees gives
Using Giambelli’s formula \(\sigma _{a,b}=\sigma _{a}\sigma _{b}  \sigma _{a+1}\sigma _{b1}\) (Eisenbud and Harris 2016, Prop. 4.16), we reduce to computing \(\deg ([Z] \cdot \sigma _{a}\sigma _{b})\) for \(0\le b\le \left\lfloor \frac{\dim Z}{2}\right\rfloor \). By Kleiman transversality, we have
where H and \(H'\) are general linear subspaces of \(\left{\mathscr {O}(d)}\right\) of dimension \(Na2\) and \(Nb2\), respectively.
To a point \(p = g_p h_p \in Q\) with \(g_p \in \left{\mathscr {O}(1)}\right\) and \(h_p \in \left{\mathscr {O}(d1)}\right\), associate a closed reduced subscheme \(\varLambda _p\subset Q\) containing p as follows. If \(h_p\) is irreducible, let \(\varLambda _p\) be the image of the linear embedding \(\left{\mathscr {O}(1)}\right\times \{h_p\} \rightarrow \left{\mathscr {O}(d)}\right\) given by \(g \mapsto g h_p\).
If \(h_p\) is reducible, define the subscheme \(\varLambda _p\) as the union \(\bigcup _h {\text {im}}(\left{\mathscr {O}(1)}\right \times \{h\}\rightarrow \left{\mathscr {O}(d)}\right)\), where h ranges over the (up to multiplication by units) finitely many divisors of p of degree \(d1\).
Note that for all points p, the spaces \({\text {im}}(\left{\mathscr {O}(1)}\right \times \{h\}\rightarrow \left{\mathscr {O}(d)}\right)\) meet exactly at p.
By the definition of Z, all lines \(T\in Z\) lie in Q. Furthermore, if T meets the point p, then \(T\subseteq \varLambda _p\). For \(H\subseteq \left{\mathscr {O}(d)}\right\) a linear subspace of dimension \(Na2\), define \(Q':=H\cap Q\). For general H, the subscheme \(Q'\) is a smooth subvariety of dimension \(bn+1\) such that for a general point \(p=gh\) of \(Q'\) with \(h\in \left{\mathscr {O}(d)}\right\), the polynomial h is irreducible.
Next, we consider the case \(n=2\) or \(\dim Z\) odd.
Claim
For genereal H, for each point \(p\in Q'\) we have \(\varLambda _p \cap H =\{p\}\).
Proof
(of Claim) Let \(\mathscr {H}\) denote the Grassmannian \({\text {Gr}}(\dim H+1, N)\). Define the closed subset \(X\subseteq Q\times \mathscr {H}\) by
The fibers of the induced map \(X\rightarrow \mathscr {H}\) have dimension at least one. Hence, to prove that the desired condition on H is an open condition, it suffices to prove \(\dim (X) \le \dim (\mathscr {H})\).
The fiber of the map \(X\rightarrow Q\) over a point p consists of the union of finitely many closed subsets of the form \(X'_p = \{H\in \mathscr {H} : \dim (H\cap \varLambda '_p)\ge 1\}\), where \(\varLambda '_p\simeq \mathbb {P}^n\subseteq \left{\mathscr {O}(d)}\right\) is one of the components of \(\varLambda _p\). The space \(X'_p\) is a Schubert cycle
with \(H_{n+1}\) an \((n+1)\)dimensional subspace of \(H^0(\mathscr {O}(d))\). The codimension of the cycle is \(2(\dim Q  b)\), hence also \({{\mathrm{codim}}}(X_p) = 2(\dim Q b)\). Finally, we have \(\dim (\mathscr {H})\dim (X) = {{\mathrm{codim}}}(X_p)  \dim (Q) = \dim Q  2b\).
If \(\dim Z\) is odd, then \(\dim Q  2b \ge \dim Q  \dim Z + 1 = 3n\ge 0\). If \(n=2\), we instead estimate \(\dim Q  2b \ge \dim Q  \dim Z = 2n\ge 0\). \(\square \)
Next, let
and
By the choice of H, the map \(f^{1}(Q')\rightarrow Q'\) is birational and the map \(f^{1}(Q')\rightarrow \mathrm {pr}_2f^{1}(Q')\) is even bijective. It follows that \(\varLambda ''\) and hence \(\varLambda \) have dimension \(b+1\).
The intersection of \(\varLambda \) with a general linear subspace \(H'\) of dimension \(Nb2\) is a finite set of points. For each point \(p\in Q'\), the linear subspace \(H'\) intersects each component \(\varLambda '_p\) of \(\varLambda _p\) in at most one point. For each point \(p'\in H'\cap \varLambda \) there exists a unique p such that \(p'\in \varLambda _p\).
The only line \(T\in Z\) meeting both p and \(H'\) is the one through p and \(p'\). If the intersection \(H'\cap \varLambda _p\) is empty, then there will be no line meeting p and \(H'\). Hence, \(\deg ([Z]\cdot \sigma _{a}\sigma _{b})\) is the number of intersection points of \(\varLambda \) with a general \(H'\).
Finally, the preimage \(f^{1}(Q') = f^{1}(H)\) is smooth for a general H by Bertini’s Theorem. If \(\zeta \) is the class of a hyperplane section of \(\left{\mathscr {O}(d)}\right\) we have \(f^*(\zeta ) = \alpha + \beta \), where \(\alpha \) and \(\beta \) are classes of hyperplane sections of \(\left{\mathscr {O}(1)}\right\) and \(\left{\mathscr {O}(d)}\right\), respectively. Since \(\mathrm {pr}_2\) and f have degree one, we compute
Hence, by the push–pull formula:
We then use Giambelli’s formula to obtain Eq. 1.1.
In case \(n=3\) and \(\dim Z\) even, we show that for \(b=\dim Z/2\) we have \(\deg ([Z]\cdot \sigma _{b,b})=0\). In this case, the hyperplanes H and \(H'\) have the same dimension \(Nb2\).
For \(p\in Q\), the set \(\varLambda _p\) is defined as before.
Claim
for general H of dimension \(Nb2\), we have \(\dim (\varLambda _p\cap H)=1\).
Proof
(of Claim) Define as before the closed subset \(X\subseteq Q\times \mathscr {H}\) by
The generic fiber of the projection map \(\varphi :X\rightarrow \mathscr {H}\) is onedimensional, hence we have \(\dim \varphi (X) = \dim (X)1 = \dim \mathscr {H}\). The last equation holds with \(n=3\) and \(2b=\dim Z\). Hence for all \(H\in \mathscr {H}\) we have \(\dim (\varLambda _p \cap H)\ge 1\).
On the other hand, the equality \(\dim (\varLambda _p \cap H)=1\) is attained by some, and hence by a general, H. Indeed, Define the closed subset \(X\subseteq Q\times \mathscr {H}\) by
By a similar argument as before, one needs to show that \(\dim (\mathscr {H})\dim (X) + 1\ge 0\). The fiber \(X_p\) is a Schubert cycle of codimension \(3(\dim Qb+1)\). Lastly, a computation shows \(\dim (\mathscr {H})\dim ({\widetilde{X}})+1={{\mathrm{codim}}}({\widetilde{X}}_p)\dim (Q)+1=\frac{1}{2}(2\dim Q + 18  5n)\ge 0\). \(\square \)
Now, define \(\varLambda ''\) as above. We have \(\dim \varLambda '' = \dim \left{\mathscr {O}(1)}\right + \dim \mathrm {pr}_2 f^{1}(Q') = b\). Since f is generically of degree one, we still have \(\dim \varLambda '' = \varLambda \), hence \(\dim \varLambda + \dim H' = N2 < \dim \left{\mathscr {O}(d)}\right\). It follows that a generic \(H'\) does not meet any of the lines \(T\subset Z\), hence \(\sigma _{b}\sigma _{b}\cdot [Z] = 0\). \(\square \)
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Acknowledgements
Open access funding provided by Max Planck Society. This work is a condensed version of my Master’s thesis, supervised by Daniel Huybrechts. I would like to take the opportunity to thank him for his mentorship during the writing of the thesis, as well as for his help during the preparation of this article.
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Marigliano, O. Verlinde bundles of families of hypersurfaces and their jumping lines. Beitr Algebra Geom 60, 437–444 (2019). https://doi.org/10.1007/s1336601804235
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DOI: https://doi.org/10.1007/s1336601804235