EFFECTIVE CONE OF THE BLOW UP OF THE SYMMETRIC PRODUCT OF A CURVE ANTONIO LAFACE AND LUCA UGAGLIA

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EFFECTIVE CONE OF THE BLOW UP OF THE SYMMETRIC

PRODUCT OF A CURVE

ANTONIO LAFACE AND LUCA UGAGLIA

Abstract. Let Cbe a smooth curve of genus g≥1 and let C(2)be its second

symmetric product. In this note we prove that if Cis very general, then the

blow-up of C(2)at a very general point has non-polyhedral pseudo-eﬀective

cone. The strategy is to consider ﬁrst the case of hyperelliptic curves and then

to show that having polyhedral pseudo-eﬀective cone is a closed property for

families of surfaces.

Introduction

The study of the eﬀective cone of the blow up ˜

Sof a projective surface Sat

a smooth point x∈Sis connected with the calculation of Seshadri constants.

Deciding when the (pseudo)eﬀective cone of ˜

Sis polyhedral is an open problem

even when Sis a toric surface. For instance, if the eﬀective cone of the blow up

of the weighted projective plane P(a, b, c)at a general point is not closed, then

Nagata’s Conjecture holds for abc points in P2, see [4] and [7–10] for recent results

on blow ups of weighted projective planes. In [2] it has been shown that there

exist toric surfaces whose blow up at a general point has non-polyhedral pseudo-

eﬀective cone. This result allows one to deduce that the pseudo-eﬀective cone of

the Grothendieck-Knudsen moduli space ¯

M0,n is not polyhedral for n≥10.

In this paper we focus on the second symmetric product C(2)of a positive genus

curve C. In general it is not known if the eﬀective cone of these surfaces is open.

This would be true if the Nagata Conjecture holds, as shown in [3]. Our interest is

in the blow up ˜

C(2)at a very general point p⊕q∈C(2).

Theorem 1. Let Cbe a very general curve of genus g≥1. Then the blow-up of the

symmetric product C(2)at a very general point has non-polyhedral pseudo-eﬀective

cone.

In order to prove the theorem we ﬁrst show, in Proposition 1.3, that having

polyhedral pseudo-eﬀective cone is a closed property for families of surfaces and

then we prove the following.

Theorem 2. Let Cbe a genus g≥1hyperelliptic curve with hyperelliptic involution

σ, let p∈Cand let ˜

C(2)be the blow-up of C(2)at p⊕σ(p). If the class of σ(p)−p

is non-torsion in Pic0(C)then Eﬀ(˜

C(2))is non-polyhedral.

When Cis an elliptic curve, its symmetric product is the Atyiah surface. In

this case in [6] it has been proved that if q−pis non-torsion, then ˜

C(2)contains

Date: October 24, 2022.

2010 Mathematics Subject Classiﬁcation. Primary 14M25; Secondary 14C20.

Both authors have been partially supported by Proyecto FONDECYT Regular n. 1190777.

arXiv:2210.11829v1 [math.AG] 21 Oct 2022

2 A. LAFACE AND L. UGAGLIA

inﬁnitely many negative curves. Therefore the pseudo-eﬀective cone of ˜

C(2)is not

polyhedral, and in [13] it is proved that the classes of the above mentioned curves

(together with other two classes) indeed generate the pseudo-eﬀective cone.

Our proof of Theorem 2focuses on the quotient surface ˜

Xby the action of the

hyperelliptic involution on both factors. We show that there is an irreducible curve

Bon ˜

Xhaving self intersection B2=0, whose class spans an extremal ray of the

pseudo-eﬀective cone of ˜

X, so that the latter cannot be polyhedral by [2, Proposition

2.3]. We then apply Proposition 1.1 to the double cover ˜

C(2)→˜

Xto conclude that

the pseudo-eﬀective cone of ˜

C(2)is not polyhedral.

The paper is structured as follows. In Section 1we recall some deﬁnitions and

we prove some preliminary results about the eﬀective cone of projective surfaces. In

Section 2we study the symmetric product C(2)of a curve, with particular emphasis

on the case Chyperelliptic. Section 3is devoted to the proof of Theorem 1and 2,

while in Section 4we prove some results in case g(C)=1.

Ackowledgements. It is a pleasure to thank Jenia Tevelev for several interest-

ing discussions on the subject of this paper.

1. Preliminaries

Let kbe an algebraically closed ﬁeld of arbitrary characteristic. We recall some

deﬁnitions (see for example [11,12]). If Xis a normal projective irreducible variety

over k, let Cl(X)be the divisor class group and let Pic(X)be the Picard group

of X. As usual, we denote by ∼the linear equivalence of divisors and by ≡the

numerical equivalence. Recall that for Cartier divisors D1,D2, we have D1≡D2if

and only if D1⋅C=D2⋅C, for any curve C⊆X. We let

N1(X)∶=Pic(X)/≡

be the N´eron-Severi group, i.e. the group of numerical equivalence classes of Cartier

divisors on X. We denote by ρ(X)the rank of N1(X)and by N1(X)R=N1(X)⊗ZR,

N1(X)Q=N1(X)⊗ZQ. We deﬁne the pseudo-eﬀective cone

Eﬀ(X)⊆N1(X)R,

as the closure of the eﬀective cone Eﬀ(X), i.e., the convex cone generated by

numerical classes of eﬀective Cartier divisors ([12, Def. 2.2.25]). We let Nef(X)⊆

N1(X)Rbe the cone generated by the classes of nef divisors.

Proposition 1.1. Let f∶X→Ybe a ﬁnite surjective morphism of normal Q-

factorial projective varieties. If %(X)=%(Y), then f∗∶N1(X)R→N1(Y)Ris an

isomorphism such that f∗(Eﬀ(X))=Eﬀ(Y).

Proof. Since Yis Q-factorial, the image of Pic(Y)in the N´eron-Severi group N1(Y)

has ﬁnite index. Over this subgroup the pullback is deﬁned and the projection

formula gives f∗○f∗=n⋅id, where n=deg(f). This, together with the hypothesis

ρ(X)=ρ(Y), imply that f∗∶N1(X)R→N1(Y)Ris an isomorphism whose inverse

is 1

nf∗. Then one concludes by the inclusions

f∗(Eﬀ(X))⊆Eﬀ(Y)and f∗(Eﬀ(Y))⊆Eﬀ(X).



EFFECTIVE CONE OF THE BLOW UP OF THE SYMMETRIC PRODUCT OF A CURVE 3

Proposition 1.2. Let Xbe a normal Q-factorial algebraic surface with %(X)≥3

and positive light cone Q⊆N1(X)R. Let C1,...,Cnbe irreducible curves of X.

Then the following are equivalent:

(1) Q⊆Cone([C1],...,[Cn]);

(2) Eﬀ(X)=Cone([C1],...,[Cn]).

Moreover if Eﬀ(X)is polyhedral then Eﬀ(X)=Eﬀ(X)holds and both cones are

generated by classes of negative curves.

Proof. We prove (1)⇒(2). Let [D]be a divisor class which generates an extremal

ray of the eﬀective cone. Then D2<0 so that the hyperplane D⊥intersects Q

along its interior. As a consequence at least one of the Cisatisﬁes D⋅Ci<0. Thus

any eﬀective multiple of Dcontains Ciinto its support, so that [D]=[Ci]up to

multiples.

The implication (2)⇒(1)is obvious. 

Proposition 1.3. Let X→Bbe a ﬂat projective morphism of Noetherian schemes,

whose general ﬁber is a normal Q-factorial surface with Picard lattice isometric to

the one of the special ﬁber X0over 0∈B. If the general ﬁber has polyhedral pseudo-

eﬀective cone, then the same holds for the special ﬁber.

Proof. If the Picard rank is ≤2, then the presudoeﬀective cone is polyhedral and

there is nothing to prove. We then assume that the Picard rank is at least 3. By

Proposition 1.2 the pseudo-eﬀective cone of the general ﬁber is generated by ﬁnitely

many classes of negative curves C1,...,Cn. By semicontinuity of cohomology di-

mension, each such curve Cidegenerate to a, possibly reducible, curve of X0. Let

Ci1,...,Ciribe the irreducible components of the degenerate curve. We claim that

in the N´eron-Severi space of the special ﬁber X0the following inclusions of cones

hold

Q⊆Cone([Ci]∶1≤i≤n)⊆Cone([Cij ]∶1≤i≤n, 1≤j≤ri).

Indeed, by Proposition 1.2, the ﬁrst inclusion holds true in the N´eron-Severi space

of the general ﬁber and, by the assumption on the Picard lattice of the special ﬁber,

it holds as well on the N´eron-Severi space of the special ﬁber. The second inclusion

follows by the deﬁnition of the curves Cij . Then, again by Proposition 1.2, one

concludes that Eﬀ(X0)=Cone([Cij ]∶1≤i≤n, 1≤j≤ri).

2. Symmetric product of a curve

Given a genus g≥1 curve C, we denote by C(2)its second symmetric product,

that is the quotient of C×Cby the involution τ, deﬁned by (p, q)↦(q, p), and we

denote by p⊕q∈C(2)the class of (p, q)∈C×C.

From now on we assume that Cis hyperelliptic, we ﬁx a hyperelliptic involution

σand we denote by p1,...,p2g+2∈Cits ﬁxed points. Observe that σinduces two

commuting involutions σ1, σ2on C×C, each of which acts only on one coordinate.

The group G∶=⟨σ1, σ2, τ⟩is isomorphic to D4, with center generated by the com-

position σ1⋅σ2, that we still denote by σwith abuse of notation. We have the

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EFFECTIVECONEOFTHEBLOWUPOFTHESYMMETRICPRODUCTOFACURVEANTONIOLAFACEANDLUCAUGAGLIAAbstract.LetCbeasmoothcurveofgenusgC1andletC2beitssecondsymmetricproduct.InthisnoteweprovethatifCisverygeneral,thentheblow-upofC2ataverygeneralpointhasnon-polyhedralpseudo-eectivecone.Thestrategyistoconsiderrstthec...

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EFFECTIVE CONE OF THE BLOW UP OF THE SYMMETRIC PRODUCT OF A CURVE ANTONIO LAFACE AND LUCA UGAGLIA

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