On primitive Ulrich bundles over a few projective varieties with Picard number two
Locally free sheaves without intermediate cohomology on projective varieties are said to be arithmetically Cohen-Macaulay (aCM), additionally those that attain the maximum possible number of minimal generators are called Ulrich bundles. In this paper, the author introduces the notion of primitive Ulrich bundles i.e. bundles that are extensions of direct sums of Ulrich line bundles. Let \(X\) be a smooth projective variety, a vector bundle \(E\) on \(X\) is a primitive Ulrich bundle if it arises from an exact sequence \N\[\N0\longrightarrow A\longrightarrow E\longrightarrow B\longrightarrow 0\N\]\Nwhere \(A\) and \(B\) are direct sums of Ulrich line bundles.\N\NThe author gives cohomological characterizations of primitive Ulrich bundles on the degree 6 flag threefold.\NThat is, let \(\mathscr{V}\) be an Ulrich vector bundle on \(F\subseteq\mathbb{P}^7\), a del Pezzo threefold of degree 6 and Picard number 2, such that \(h^2(\mathscr{V}(-2,-2)\otimes\mathscr{G}_1\otimes\mathscr{G}_2) = 0\) then \(\mathscr{V}\) is primitive and arises from an exact sequence\N\[\N0\longrightarrow\mathcal O_F(0,2)^{\oplus a}\longrightarrow\mathcal O_F(2,0)^{\oplus b}\longrightarrow\N\]\Nwhere\N\(\mathscr{G}_i=p_i^*\Omega^1_{\mathbb{P}^2}(h_i)=p_i^*[\Omega^1_{\mathbb{P}^2}(1)]\) and \(h_i\) generators of the Picard group of \(F\) for \(i=1,2\).\N\NThe author also gives cohomological characterizations of primitive Ulrich bundles on rational normal scrolls. That is,\Nlet \(\mathscr{V}\) be an Ulrich vector bundle on a smooth rational normal scroll, \(S\) such that \(h^i(\mathscr{V}(-i,-i-1)) = 0\) for any \(i=1,\cdots,n-1\).\NThen \(\mathscr{V}\) is primitive and arises from an exact sequence\N\[\N0\longrightarrow\mathcal O_S(0,1)^{\oplus a}\longrightarrow\mathcal O_S(c-1,0)^{\oplus b}\longrightarrow 0\N\]\NFinally, the author proposes a few open problems, concerning cohomological conditions necessary to characterize primitive Ulrich bundles on other varieties \(X\) with different degree and Picard number.
scientific article; zbMATH DE number 7966906