AbstractWe introduce the notion of rooftop flip, namely a small modification among normal projective varieties which is modeled by a smooth projective variety of Picard number 2 admitting two projective bundle structures. Examples include the Atiyah flop and the Mukai flop, modeled respectively by $$\mathbb {P}^1\times \mathbb {P}^1$$ P 1 × P 1 and by $$\mathbb {P}\left( T_{\mathbb {P}^2}\right) $$ P T P 2 . Using the Morelli-Włodarczyk cobordism, we prove that any smooth projective variety of Picard number 1, endowed with a $${\mathbb C}^*$$ C ∗ -action with only two fixed point components, induces a rooftop flip.

AbstractThe v-number of a graded ideal is an algebraic invariant introduced by Cooper et al., and originally motivated by problems in algebraic coding theory. In this paper we study the case of binomial edge ideals and we establish a significant connection between their v-numbers and the concept of connected domination in graphs. More specifically, we prove that the localization of the v-number at one of the minimal primes of the binomial edge ideal $$J_G$$ J G of a graph G coincides with the connected domination number of the defining graph, providing a first algebraic description of the connected domination number. As an immediate corollary, we obtain a sharp combinatorial upper bound for the v-number of binomial edge ideals of graphs. Lastly, building on some known results on edge ideals, we analyse how the v-number of $$J_G$$ J G behaves under Gröbner degeneration when G is a closed graph.

AbstractLet X be any smooth prime Fano threefold of degree $$2g-2$$ 2 g - 2 in $${\mathbb P}^{g+1}$$ P g + 1 , with $$g \in \{3,\ldots ,10,12\}$$ g ∈ { 3 , … , 10 , 12 } . We prove that for any integer d satisfying $$\left\lfloor \frac{g+3}{2} \right\rfloor \leqslant d \leqslant g+3$$ g + 3 2 ⩽ d ⩽ g + 3 the Hilbert scheme parametrizing smooth irreducible elliptic curves of degree d in X is nonempty and has a component of dimension d, which is furthermore reduced except for the case when $$(g,d)=(4,3)$$ ( g , d ) = ( 4 , 3 ) and X is contained in a singular quadric. Consequently, we deduce that the moduli space of rank–two slope–stable ACM bundles $${\mathcal F}_d$$ F d on X such that $$\det ({\mathcal F}_d)={\mathcal O}_X(1)$$ det ( F d ) = O X ( 1 ) , $$c_2({\mathcal F}_d)\cdot {\mathcal O}_X(1)=d$$ c 2 ( F d ) · O X ( 1 ) = d and $$h^0({\mathcal F}_d(-1))=0$$ h 0 ( F d ( - 1 ) ) = 0 is nonempty and has a component of dimension $$2d-g-2$$ 2 d - g - 2 , which is furthermore reduced except for the case when $$(g,d)=(4,3)$$ ( g , d ) = ( 4 , 3 ) and X is contained in a singular quadric. This completes the classification of rank–two ACM bundles on prime Fano threefolds. Secondly, we prove that for every $$h \in {\mathbb Z}^+$$ h ∈ Z + the moduli space of stable Ulrich bundles $${\mathcal E}$$ E of rank 2h and determinant $${\mathcal O}_X(3h)$$ O X ( 3 h ) on X is nonempty and has a reduced component of dimension $$h^2(g+3)+1$$ h 2 ( g + 3 ) + 1 ; this result is optimal in the sense that there are no other Ulrich bundles occurring on X. This in particular shows that any prime Fano threefold is Ulrich wild.

AbstractWe prove that a map germ $$f:(\mathbb {C}^n,S)\rightarrow (\mathbb {C}^{n+1},0)$$ f : ( C n , S ) → ( C n + 1 , 0 ) with isolated instability is stable if and only if $$\mu _I(f)=0$$ μ I ( f ) = 0 , where $$\mu _I(f)$$ μ I ( f ) is the image Milnor number defined by Mond. In a previous paper we proved this result with the additional assumption that f has corank one. The proof here is also valid for corank $$\ge 2$$ ≥ 2 , provided that $$(n,n+1)$$ ( n , n + 1 ) are nice dimensions in Mather’s sense (so $$\mu _I(f)$$ μ I ( f ) is well defined). Our result can be seen as a weak version of a conjecture by Mond, which says that the $$\mathscr {A}_e$$ A e -codimension of f is $$\le \mu _I(f)$$ ≤ μ I ( f ) , with equality if f is weighted homogeneous. As an application, we deduce that the bifurcation set of a versal unfolding of f is a hypersurface.

We describe a combinatorial condition on a graphwhich guarantees that all powers of its vertex cover ideal are componentwise linear. Then motivated by Eagon and Reiner’s Theorem we study whether all powers of the vertex cover ideal of a Cohen-Macaulay graph have linear free resolutions. After giving a complete characterization of Cohen-Macaulay cactus graphs (i.e., connected graphs in which each edge belongs to at most one cycle) we show that all powers of their vertex cover ideals have linear resolutions.