arXiv subjects
Publications and source records attributed to Nick Rekuski.
Fix a polarised Calabi-Yau threefold $(X,H)$. We reduce a version of the Bayer-Macr\`i-Toda conjecture for $(X,H)$, which ensures the existence of Bridgeland stability conditions on $X$, to verifying a Brill-Noether-type inequality for curves on $X$. We then prove this inequality for a broad class of Calabi-Yau threefolds, including complete intersection Calabi-Yau threefolds in weighted projective spaces.
Using a recent description of the geometric stability manifold, we show the geometric stability manifold associated to any smooth projective complex surface is contractible. We then use this result to demonstrate infinitely many new families of surfaces whose stability manifold is contractible.
We consider the question: which elliptic curves appear as the Jacobian of a smooth curve of genus one splitting a Severi--Brauer variety? We provide three new examples. First, we show that if $E$ is any elliptic curve over an algebraically closed field $k$ and if $F/k$ is a perfect field extension, then there exists a principal homogeneous space for $E_F$ splitting a Severi--Brauer variety $X=\mathrm{SB}(A)$ over $F$ if and only if $A$ is Brauer equivalent to a cyclic algebra. Along the way, we also give a uniform proof of a generalization of results due to O'Neil, Clark and Sharif, and Antieau and Auel. Second, we give an example of an elliptic curve $E$ over a field $k$ together with a central simple algebra $A/k$ of degree $4$ such that $E$ is the Jacobian of a smooth genus one curve $C$ embedded in the Severi--Brauer variety $X=\mathrm{SB}(A)$ as a degree 8 curve and such that $E$ is not the Jacobian of any genus one curve of smaller degree contained in $X$. Our example is, in some sense, as small as possible in both dimension and arithmetic complexity. Third, we show that for any odd integer $n\geq 3$ there is a central simple algebra $A$ of degree $n$ over a local field $k$ which is split by the principal homogeneous space of an elliptic curve $E/k$ but not by any principal homogeneous space for any quadratic twist of $E$. This generalizes a recent result of Saltman in the case of surfaces to arbitrary even dimension.
Using techniques from Bridgeland stability, we show the kernel sheaf associated to sufficiently positive Gieseker stable sheaf on a Del Pezzo surface is slope stable. This is the first effective stability result for kernel sheaves associated to higher rank sheaves on surfaces and the first stability result for kernel sheaves associated to Gieseker stable sheaves.
We compute the field of rational local unitary invariants for locally maximally mixed states and symmetrically mixed states of two qubits. In both cases, we prove that the field of rational invariants is purely transcendental. We also construct explicit geometric quotients and prove that they are always rational. All the results are obtained by working over the field of real numbers, employing methods from classical and geometric invariant theory over arbitrary fields of characteristic zero.
We calculate the field of rational local unitary invariants for mixed states of two qubits, by employing methods from algebraic geometry. We prove that this field is rational (i.e. purely transcendental), and that it is generated by nine algebraically independent polynomial invariants. We do so by constructing a relative section, in the sense of invariant theory, whose Weyl group is a finite abelian group. From this construction, we are able to give explicit expressions for the generating invariants in terms of the Bloch matrix representation of mixed states of two qubits. We also prove similar rationality statements for the local unitary invariants of symmetric mixed states of two qubits. Our results apply to both complex-valued and real-valued invariants.
We show the kernel sheaf associated to a sufficiently positive torsion-free sheaf of rank 1 is slope stable. Furthermore, we are able to give an explicit bound for "sufficiently positive." This settles a conjecture of Ein-Lazarsfeld-Mustopa. The main technical lemma is a bound on the number of global sections of a torsion-free, globally generated sheaf in terms of its rank, degree, and invariants of the variety.