This is a review of a paper originally written in Russian:
	http://www.mi.ras.ru/~galkin/papers/2d.pdf

A short abstract: We study weak LG-models mirror symmetric to del Pezzo surfaces. They are shown to exist, and if the surface is not S_8 or S_7 fiberwise compactifications of these models are exactly
the pencils of elliptic curves of Euler characteristic 0 (i.e. 4 semisiple singular fibers or 2 semisimple and one nonsemisimple). Also we illustrate the Dubrovin's conjecture relating
elliptic pencils of Euler characteristic 0 and 3-blocked full exceptional collections.

A short review. The paper consists of 5 sections.

In 1 section we fix the notations for elliptic surfaces:
Kodaira-Neron and Weierstrass models,
Kodaira's classification of singular fibers/monodromies (1.1),
Shioda modular elliptic surfaces,
6 Beauville's surfaces (1.4),
6 Miranda--Persson's extremal surfaces (1.6).

Section 2 is used to fix the notations of Gromov-Witten theory:
GW invariants
I-series and restricted I-series (2.1),
quantum Lefshetz principle (2.2),
quantum cohomologies (2.4),
spectrum of variety (i.e. characteristic polynomial of * c_1(X); 2.5),
quantum minimality (2.7),
Givental's computation of I-series for c.i. in toric varieties (2.8),
corollary of 2.8 - fundamental term or I-series is equal to
series of constant terms of the potential (Laurent polynomial) (2.9),
I-series of Grassmanian (2.10),
I-series of del Pezzo surfaces (2.11-2.20).

Section 3 is the definition of weak LG model:
series of constant terms Phi_f(t) of Laurent polynomial f (3.1),
Phi_f is Pichard-Fuchs period of 1/f (3.2),
Definition. f is weak LG of X if Phi_f(t) = I_X(t) and fibers of f are CY (3.3),
support and Newton polytope of Laurent polynomial (3.4),
nondegeneracy by Kushnirenko-Danilov-Khovansky (3.5),
properties of nondegeneracy (3.6),
Danilov-Khovansky's calculation of MHS on nondegenerate hypersurfaces in tori (3.7)

Section 4. LG for del Pezzo surfaces.
List of 16 toric del Pezzo with du Val singularities (4.1),
du Val's description of del Pezzo surfaces of degree >2 with du Val singularities (4.3),
surfaces from 4.2 are degenerations of smooth del Pezzo surfaces (4.4),
-//-                 4.1           -//-                                                         (4.5),
action of G_a and G_m^2 on the space of Laurent polynomials (4.6),
divisorial-monomial correspondence,
example of projective plane (4.7),
del Pezzo surfaces except S_8 and S_7 are quantum minimal (4.8),
list of weak LG of toric del Pezzo constructed from the degenerations to du Val toric del Pezzos (4.9),
isogenies (4.10), base changes (4.11)
[the list was constructed by finding the possible Laurent polynomials with given Newton polytope
and minimal ramification (Euler characteristic 0)]
j-invariants are Belyi functions (4.12)
different weak LG-models corresponding to different degenerations
are fiberwise birationally symplectomorphic (4.17)
[done by direct computation, but later I found a nice visual proof by elementary "cluster mutations"],
end of the proof of 4.9 (4.17-4.21).
Subsection 4.1 - Hori-Vafa models:
HV for S_1, S_2, S_3, S_4 (4.22),
4.22 is Miranda-Persson's list (4.23),
example for s_4 (4.24).

Section 5: Monodromy.
derived category, full exceptional collections, euler form, antisymmetrized euler form,
symplectic reflections
Dubrovin's conjecture (local system L over the punctured line
 constructed from exceptional collection in derived category coincides with R^1 w_* \Z) (5.1),
blocked exceptional collections,
Karpov-Nogin's description of 3-blocked exceptional collections on del pezzos (5.2-5.7),
corollary - main formula for monodromies of L:
            I_e^a I_f^b I_g^c I_h^d = 1                               (5.8),
5.8 is exactly the type of relations one uses to construct topological elliptic surfaces
via string junctions, and (5.8) is corresponding to Beauville's surfaces.