On the Quantum Cohomology of some Fano threefolds and a conjecture of Dubrovin

We obtain a classification of a Q-factorial Q-Fano 3-fold $X$ with the following properties: the Picard number of $X$ is 1; the Gorenstein index of $X$ is 2; the Fano index of $X$ is 1/2; $h^0 (-K_X) \geq 4$; there exists an index 2 point $P$ such that $(X,P)\simeq (\{xy +f(z^2,u)=0\} / \Bbb Z_2 (1,1,1,0), o)$ with $ord f(Z,U)=1$.

We describe a method for counting maps of curves of given genus (and variable moduli) to $\Bbb P^2$, essentially by splitting the $\Bbb P^2$ in two; then specialising to the case of genus 0 we show that the method of quantum cohomology may be viewed as the 'mirror' of the former method where one splits the $\Bbb P^1$ rather than the $\Bbb P^2$, and we indicate a proof of the associativity of quantum multiplication based on this idea.

For even dimensional smooth complete intersections, of dimension at least 4, of two quadric hypersurfaces in a projective space, we study the genus zero Gromov-Witten invariants by the monodromy group of its whole family. We compute the invariants of length 4 and show that, besides a special invariant, all genus zero Gromov-Witten invariants can be reconstructed from the invariants of length 4. In dimension 4, we compute the special invariant by solving a curve counting probl...

Quantum K-theory of a smooth projective variety at genus zero is a collection of integers that can be assembled into a generating series $J(Q,q,t)$ that satisfies a system of linear differential equations with respect to $t$ and $q$-difference equations with respect to $Q$. With some mild assumptions on the variety, it is known that the full theory can be reconstructed from its small $J$-function $J(Q,q,0)$ which, in the case of Fano manifolds, is a vector-valued $q$-hypergeo...

This article constructs a smooth weak Fano threefold of Picard number two with small anti-canonical morphism that arises as a blowup of a smooth curve of genus 5 and degree 8 in $\mathbb{P}^3$. While the existence of this weak Fano was known as a numerical possibility in \cite{CM13} and constructed in BL12, this paper removes the dependencies on the results in \cite{JPR11} needed in the construction of BL12 and constructs the link in the style of ACM17.

We prove in the case of minimal Fano threefolds a conjecture stated by Dubrovin at the ICM 1998 in Berlin. The conjecture predicts that the symmetrized/alternated Euler characteristic pairing on $K_0$ of a Fano variety with an exceptional collection expressed in the basis of the classes of the exceptional objects coincides with the intersection pairing of the vanishing cycles in Dubrovin's second connection. We show that the conjecture holds for $V_{22}$, a minimal Fano three...

We generalize Givental's Theorem for complete intersections in smooth toric varieties in the Fano case. In particular, we find Gromov--Witten invariants of Fano varieties of dimension $\geq 3$, which are complete intersections in weighted projective spaces and singular toric varieties. We generalize the Riemann-Roch equations to weighted projective spaces. We compute counting matrices of smooth Fano threefolds with Picard group Z and anticanonical degrees 2, 8, and 16.

The dth symmetric product of a curve of genus g is a smooth projective variety. This paper is concerned with the little quantum cohomology ring of this variety, that is, the ring having its 3-point Gromov-Witten invariants as structure constants. This is of considerable interest, for example as the base ring of the quantum category in Seiberg-Witten theory. The main results give an explicit, general formula for the quantum product in this ring unless d is in the narrow interv...

We classified prime $\mathbb{Q}$-Fano $3$-folds $X$ with only $1/2(1,1,1)$-singularities and with $h^{0}(-K_{X})\geq 4$ a long time ago. The classification was undertaken by blowing up each $X$ at one $1/2(1,1,1)$-singularity and constructing a Sarkisov link. The purpose of this paper is to reveal the geometries behind the Sarkisov links for $X$ in 5 classes. The main result asserts that any $X$ in the 5 classes can be embedded as linear sections into bigger dimensional $\mat...

This paper was written in 1982. Ideas and methods of "Clemens C.H., Griffiths Ph. The intermediate Jacobian of a cubic threefold" are applied to a Fano threefold X of genus 6 -- intersection of Grassmann sixfold with two hyperplanes and a quadric. We prove: 1. The Fano surface F(X) of X is smooth and irreducible. Hodge numbers and some other invariants of F(X) are calculated. 2. Tangent bundle theorem for X, and its geometric interpretation. It is shown that F(X) defines ...