On the equations for universal torsors over del Pezzo surfaces

Abstract

We describe equations of the universal torsors over del Pezzo surfaces of degrees from 2 to 5 over an algebraically closed field in terms of the equations of the corresponding homogeneous space G/P. We also give a generalization for fields that are not algebraically closed.

Picard-graded Betti numbers and the defining ideals of Cox rings

“…Considering all such pairs in X r we immediately obtain Table 2 which contains the Picard degrees of the minimal generators of I r with anticanonical degree 2. Serganova and Skorobogatov show (see Theorem 2.5 in [14]) that certain quadratic equations in the above multidegrees suffice to define the ideal I X up to radical. Their work extends some of their previous results [13] and the work of Derenthal [7] on embedding the universal torsors of Del Pezzo surfaces in homogeneous spaces.…”

“…A proof of this conjecture for Del Pezzo surfaces of degree at least four and for generic surfaces of degree at least three appears in [15]. It is shown in [14] that the conjecture holds for surfaces of degree at least two up to radical. Besides generalizing these results, the approach in this paper proves that quadratic generation of the Cox rings of Del Pezzo surfaces depends only on numerical properties.…”

The maximum cut problem on blow-ups of multiprojective spaces

“…The varieties X a,b,c have been the focus of much recent work by Mukai, Castravet-Tevelev, Serganova-Skorobogatov, Sturmfels, Xu and the second author among others. They have appeared in connection to Mukai's answer to Hilbert's 14-th problem [11], [2], have been studied because of their close relationship with homogeneous spaces [9], [10], [16] and because of their remarkable combinatorial commutative algebra [15].…”

Blow-ups of ${\bf P}^{n-3}$ at $n$ points and spinor varieties

“…A key insight of Serganova and Skorobogatov [14] is that the universal torsor is recovered by intersecting several torus translates of the corresponding homogeneous space. The chosen elements in the torus are determined by the moduli of the surface.…”