An intersection bound for rank 1 loci, with applications to Castelnuovo and Clifford theory.

*(English)*Zbl 0798.14029Subvarieties of a projective space defined by the vanishing of minors of a matrix, or more generally subvarieties defined as the degeneracy loci of maps \(\varphi:{\mathcal V}\to {\mathcal W}\) between vector bundles on (say) a smooth variety \(X\), are ubiquitous in algebraic geometry. In the case when such a rank \(k\) locus \(Y\) has the “expected” codimension \((\text{rank} {\mathcal V}-k)\) \((\text{rank} {\mathcal W}-k)\) there are familiar methods, such as Porteous’ formula, that give information about \(Y\) in terms of \({\mathcal V}\) and \({\mathcal W}\).

In this paper we focus on the rank 1 loci of 1-generic matrices (or matrices that satisfy a slightly weaker nondegeneracy condition defined in §1) and how they intersect nondegenerate curves in the projective space. Our main technical results, presented in §1 below, are of the form: If \(X\) is a variety defined (scheme-theoretically) in \(\mathbb{P}^ n\) by the vanishing of the \(2 \times 2\) minors of a suitably “nondegenerate” matrix \(f\) of linear forms, then the number of intersections of \(X\) with a nondegenerate smooth curve \(C\) of degree \(d\) is bounded by some number \(A\) depending on \(d\), the complexity of \(C\), and the size of the matrix \(f\).

As a first application we give, in §2, a generalization to schemes of a lemma of Castelnuovo. Our result says that there is a rational normal curve passing through any finite subscheme \(\Gamma\) of degree \(\geq 2n+3\) in \(\mathbb{P}^ n\) that is in linearly general position and imposes at most \(2n+1\) conditions on quadrics. This result allows us to imitate the theory of Castelnuovo bounding the genus of a curve in terms of its degree in the case of certain nonreduced curves. Applying the bound in the case of ribbons (double structures on smooth curves) we deduce new bounds on the degree of the normal bundle of a space curve in §3.

One group of applications of our results is to the products of linear series. Recall that a linear series \(L\) on a scheme \(X\) is a pair \(L=({\mathcal L},V)\), with \({\mathcal L}\) a line bundle on \(X\) and \(V\) a subspace of \(H^ 0 {\mathcal L}\). The dimension of \(L\) is by definition one less than the vector space dimension of \(V\). If \(L_ 1=({\mathcal L}_ 1, V_ 1)\) and \(L_ 2=({\mathcal L}_ 2, V_ 2)\) are linear series, we define the product \(L_ 1L_ 2\) to be the series \(L_ 1L_ 2=({\mathcal L}_ 1 \otimes {\mathcal L}_ 2\), Image\((V_ 1 \otimes V_ 2 \to H^ 0 ({\mathcal L}_ 1 \otimes {\mathcal L_ 2}))\). If the dimensions of \(L_ 1\) and \(L_ 2\) are both at least 1, we define the Clifford index of the pair \(L_ 1\), \(L_ 2\) to be \(\text{Cliff} (L_ 1,L_ 2)=\dim L_ 1 L_ 2-\dim L_ 1-\dim L_ 2\). A simple general argument about 1-generic pairings shows that \(\text{Cliff}(L_ 1,L_ 2) \geq 0\) for any pair of linear series. Using the techniques developed here, we are able to classify, in §4, the pairs of series of Clifford index 1. For example, on a curve, such a pair of series must come from either complete series on a curve of arithmetic genus 1 or from certain pairs of incomplete series on \(\mathbb{P}^ 1\).

Finally, in §5, we use the previous theory to give information about the rank 1 locus \(Y\) of a 1-generic \((a+1) \times (b+1)\) matrix of linear forms on \(\mathbb{P}^ n\). The case where the codimension is \(ab\), the largest possible, is the case of “expected” codimension, mentioned above.

Here we partially deal with the case where the codimension is the next larger value, \(a+b\).

In this paper we focus on the rank 1 loci of 1-generic matrices (or matrices that satisfy a slightly weaker nondegeneracy condition defined in §1) and how they intersect nondegenerate curves in the projective space. Our main technical results, presented in §1 below, are of the form: If \(X\) is a variety defined (scheme-theoretically) in \(\mathbb{P}^ n\) by the vanishing of the \(2 \times 2\) minors of a suitably “nondegenerate” matrix \(f\) of linear forms, then the number of intersections of \(X\) with a nondegenerate smooth curve \(C\) of degree \(d\) is bounded by some number \(A\) depending on \(d\), the complexity of \(C\), and the size of the matrix \(f\).

As a first application we give, in §2, a generalization to schemes of a lemma of Castelnuovo. Our result says that there is a rational normal curve passing through any finite subscheme \(\Gamma\) of degree \(\geq 2n+3\) in \(\mathbb{P}^ n\) that is in linearly general position and imposes at most \(2n+1\) conditions on quadrics. This result allows us to imitate the theory of Castelnuovo bounding the genus of a curve in terms of its degree in the case of certain nonreduced curves. Applying the bound in the case of ribbons (double structures on smooth curves) we deduce new bounds on the degree of the normal bundle of a space curve in §3.

One group of applications of our results is to the products of linear series. Recall that a linear series \(L\) on a scheme \(X\) is a pair \(L=({\mathcal L},V)\), with \({\mathcal L}\) a line bundle on \(X\) and \(V\) a subspace of \(H^ 0 {\mathcal L}\). The dimension of \(L\) is by definition one less than the vector space dimension of \(V\). If \(L_ 1=({\mathcal L}_ 1, V_ 1)\) and \(L_ 2=({\mathcal L}_ 2, V_ 2)\) are linear series, we define the product \(L_ 1L_ 2\) to be the series \(L_ 1L_ 2=({\mathcal L}_ 1 \otimes {\mathcal L}_ 2\), Image\((V_ 1 \otimes V_ 2 \to H^ 0 ({\mathcal L}_ 1 \otimes {\mathcal L_ 2}))\). If the dimensions of \(L_ 1\) and \(L_ 2\) are both at least 1, we define the Clifford index of the pair \(L_ 1\), \(L_ 2\) to be \(\text{Cliff} (L_ 1,L_ 2)=\dim L_ 1 L_ 2-\dim L_ 1-\dim L_ 2\). A simple general argument about 1-generic pairings shows that \(\text{Cliff}(L_ 1,L_ 2) \geq 0\) for any pair of linear series. Using the techniques developed here, we are able to classify, in §4, the pairs of series of Clifford index 1. For example, on a curve, such a pair of series must come from either complete series on a curve of arithmetic genus 1 or from certain pairs of incomplete series on \(\mathbb{P}^ 1\).

Finally, in §5, we use the previous theory to give information about the rank 1 locus \(Y\) of a 1-generic \((a+1) \times (b+1)\) matrix of linear forms on \(\mathbb{P}^ n\). The case where the codimension is \(ab\), the largest possible, is the case of “expected” codimension, mentioned above.

Here we partially deal with the case where the codimension is the next larger value, \(a+b\).

##### MSC:

14M12 | Determinantal varieties |

14C20 | Divisors, linear systems, invertible sheaves |

14H60 | Vector bundles on curves and their moduli |

14C17 | Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry |