Stable Ulrich bundles (Q2909480)

Recall that a coherent sheaf \(E\) on an integral projective variety \((X, {\mathcal O}_X)\) is called ACM (\textit{arithmetically Cohen-Macaulay}) if \(H^i(E(t))=0\) for all \(1<i< \text{dim } X\) and all \(t\in \mathbb Z\). If \(d\) is the degree of \(X\) and \(r\) the rank of \(E\), it was shown by the authors, in a previous paper, that the graded module \(\bigoplus _t H^0(E(t))\) is generated by at most \(dr\) elements. The sheaves for which the maximal number is achieved are called \textit{Ulrich sheaves}. In this paper one calls \textit{Ulrich} those sheaves which have all these generators in degree \(0\).NEWLINENEWLINEIn this paper one studies Ulrich bundles on smooth cubic surfaces \(X \subset {\mathbb P}^3\). The main results of this paper are:NEWLINENEWLINE Theorem 1.1. If \(X\) is a divisor on a smooth cubic surface in \(\mathbb P^3\) and \(r\) is an integer \(\geq 0\), then there exist rank \(r\) stable Ulrich bundles on \(X\) iff \(0\leq D \cdot L \leq 2r\) for all lines \(L\) on \(X\) and \(D\cdot T \geq 2r\) for all twisted cubics \(T\) on \(X\), with one exception. In this case the moduli space of stable vector bundles on \(X\) of rank \(r\), \(c_1=D\) and \(c_2=\frac{D^2-r}{2}\) is irreducible, smooth, of dimension \(D^2-2r^2+1\) and consists only of stable Ulrich bundles.NEWLINENEWLINE Theorem 1.2. The moduli space of rank \(r\geq 2\) stable Ulrich bundles on a general cubic threefold \(Y \subset {\mathbb P}^4\) is nonempty, of dimension \(r^2+1\). It has an open subset for which the restriction to a hyperplane section gives an étale dominant map to the moduli space on the cubic surface.NEWLINENEWLINEThe last theorem uses essentially the results of F.~Geiss and F.-O.~Schreyer, included in an appendix to this paper. They show, using ingenuosly computations in Macaulay2 over a finite field and semicontinuity, that, if \(N_{E/X}\) is the normal bundle of a general elliptic normal curve \(E\) of degree \(5\) on a general cubic threefold \(X\subset {\mathbb P}^4\) over an algebraically closed field of characteristic \(0\), then \(N_{E/X}(-1)\cong L\oplus L^{-1}\), where \(L\in \mathrm{Pic}^0(X)\), \(L \not \cong {\mathcal O}_E\). In particular \(H^1(N_{E/X}(-1)=0\). The last vanishing is proven also for a general arithmetically Cohen-Macaulay curve \(C\) of genus \(10\) and degree \(12\) on a cubic threefold \(X\subset {\mathbb P}^4\). Then there is proven the existence of a nonempty open set of triples \((C,E,X)\) which satisfy: \(E \cap C =\emptyset \), \(\text{dim }\mathrm{Ext}^1(I_{E/X}(2), {\mathcal O}_X)=1\) and for the nontrivial extension \(F\), \(H^1(F\otimes I_{C/X})=H^2(F\otimes I_{C/X}) =0\).